text:
\section{Weil sums of binomials}\label{Wally}

A {\it Weil sum} is a finite field character sum where the character is evaluated with a polynomial argument.
We are interested in additive character Weil sums, that is, sums of the form
\[
\sum_{x \in F} \phi(f(x)),
\]
where $F$ is a finite field, $\phi\colon F \to {\mathbb C}$ is an additive character, and $f(x)$ is a polynomial in $F[x]$.
We typically use the {\it canonical additive character}, $\psi \colon F \to {\mathbb C}$ with $\psi(x)=e^{2\pi i \Tr(x)/p}$, where $\Tr \colon F \to \F_p$ is the absolute trace, because for any additive character $\phi \colon F \to {\mathbb C}$ there is some $c \in F$ such that $\phi(x)=\psi(c x)$ for all $x \in F$, so that the above sum can be recast into the form
\[
\sum_{x \in F} \psi(g(x)),
\]
where $g(x)=c f(x) \in F[x]$.
In this paper, we are specifically interested in Weil sums with binomial arguments, that is, sums of the form
\begin{equation}\label{Edgar}
\sum_{x \in F} \psi(a x^d + b x^e),
\end{equation}
where $a,b \in F^*$ and $d\not=e$.
Such sums have been studied in \cite{Mordell,Davenport-Heilbronn,Karatsuba,Carlitz-78,Carlitz-79,Coulter,Cochrane-Pinner-03,Cochrane-Pinner-11}.
For example, we see that the Kloosterman sum,
\[
\sum_{x \in F^*} \psi(a x+ b x^{-1})
\]
is one less than what one obtains in \eqref{Edgar} in the special case where $d=1$ and $e=\card{F}-2$.

We are particularly interested in the case where the exponents that appear in our binomial are invertible modulo $\card{F^*}$.
\begin{definition}[Invertible Exponent]
If $F$ is a finite field, we say that a positive integer $d$ is an {\it invertible exponent over $F$} (or just say that $d$ is {\it invertible over $F$}) to mean that $\gcd(d,\card{F^*})=1$.
\end{definition}
The map $x \mapsto x^d$ is a permutation of the finite field $F$ if and only if $d$ is invertible over $F$, and we have a special name for such permutations.
\begin{definition}[Power Permutation]\label{Jane}
If $F$ is a finite field, a {\it power permutation} is a mapping from $F$ to itself of the form $x \mapsto x^d$ for some $d$ that is invertible over $F$.
\end{definition}
If $d$ and $e$ in \eqref{Edgar} are invertible over $F$, we may use this fact to reparameterize the Weil sum by replacing $x$ with $a^{-1/d} x^{1/e}$ (where inversion in exponents is modulo $\card{F^*}$) to obtain
\[
\sum_{x \in F} \psi(x^{d/e}+b a^{-e/d} x),
\]
so it suffices to study sums in the following standard form.
\begin{definition}[$W_{F,d}$]\label{William}
If $F$ is a finite field with canonical additive character $\psi$, and $d$ is invertible over $F$, then $W_{F,d}$ always denotes the function from $F$ to ${\mathbb C}$ with
\[
W_{F,d}(a)= \sum_{x \in F} \psi(x^d - a x),
\]
for each $a \in F$.
\end{definition}
For fixed $F$ and $d$, we are interested in the values of $W_{F,d}(a)$ as $a$ runs through $F^*$.
\begin{definition}[Weil Spectrum]
If $F$ is a finite field and $d$ is an invertible exponent over $F$, then the {\it Weil spectrum} associated with $F$ and $d$ is the set $\{W_{F,d}(a): a \in F^*\}$.
\end{definition}
Readers familiar with the Walsh spectrum of a power permutation (introduced later in this paper in Section \ref{Persephone}) should note that the Walsh spectrum includes the value of $W_{F,d}(0)$ while the Weil spectrum defined here does not.
When one does consider $W_{F,d}(0)$, the polynomial in the definition of $W_{F,d}$ is a monomial, and the fact that $x \mapsto x^d$ is a permutation of $F$ makes $W_{F,d}(0)=0$.

In this survey we first review the applications of the Weil sums $W_{F,d}(a)$ to nonlinearity of finite field permutations (Section \ref{Persephone}), correlation of pseudorandom sequences (Section \ref{Martha}), and coding theory and finite geometry (Section \ref{Elijah}).
We then review their basic properties using the Galois theory of finite fields (Section \ref{Bartholomew}), the Galois theory of cyclotomic extensions of ${\mathbb Q}$ (Section \ref{Cecilia}), and power moments (Section \ref{Daria}).
Finally we survey the current state of knowledge concerning these sums, including bounds on them, both Archimedean (Section \ref{Edwin}) and non-Archimedean (Section \ref{Felix}), the number of distinct values in the Weil spectrum (Section \ref{Gerald}), and the presence or absence of zero in the Weil spectrum (Section \ref{Nigel}).
These last four sections state some of the principal open problems in the field, and indicate progress toward their resolution.

\section{Nonlinearity of power permutations}\label{Persephone}

Power permutations, introduced in Definition \ref{Jane}, are of interest in cryptography, for example, in the S-boxes of symmetric key cryptosystems.
We are interested in how far our power permutation is from being linear.
To this end, we define a transform that measures nonlinearity.
\begin{definition}[Walsh Transform]\label{Eric}
Let $F$ be a finite field of characteristic $p$ and $f\colon F \to F$.
Then the {\it Walsh transform} of $f$ is the function $W_f \colon F^2 \to {\mathbb C}$ given by
\[
W_f(b,a)=\sum_{x \in F} \psi(b f(x)-a x),
\]
where $\psi(y)=e^{2\pi i \Tr(y)/p}$ is the canonical additive character of $F$, with $\Tr$ the absolute trace of $F$.
\end{definition}
When $p=2$ we have $\psi(y)=(-1)^{\Tr(y)}$ so $W_f(b,a)$ measures the number of agreements minus the number of disagreements between the functions $x \mapsto \Tr(b f(x))$ and the ${\mathbb F}_2$-linear functions $x\mapsto \Tr(a x)$.
In other characteristics $p$, the meaning of the Walsh transform values is not as obvious, but a large magnitude value still indicates a high degree of resemblance.
Since we are trying to avoid any resemblance, we should examine all the values $W_f(b,a)$ except those with $b=0$, for $W_f(0,a)$ tells us nothing about $f$: it is trivially $\card{F}$ when $a=0$ and trivially $0$ otherwise.
\begin{definition}[Walsh Spectrum]
Let $F$ be a finite field and $f\colon F \to F$ with Walsh transform $W_f$.
The {\it Walsh spectrum of f} is the set of values $\{W_f(b, a): b \in F^* , a \in F \}$.
\end{definition}
A highly nonlinear function $f$ is one where all the values in the Walsh spectrum have small magnitude.
When $d$ is an invertible exponent over our field $F$, and $f$ is the power permutation $f(x)=x^d$, then we have Walsh transform
\begin{equation}\label{Francis}
W_f(b,a)=\sum_{x \in F} \psi(b x^d-a x).
\end{equation}
When $b\not=0$, the reparameterization described in Section \ref{Wally} shows that $W_f(b,a)=W_{F,d}(b^{-1/d} a)$.
Thus we see that the Walsh spectrum for our power permutation is given by the values of $W_{F,d}(a)$ as $a$ runs through $F$, that is, the Walsh spectrum is just the Weil spectrum along with $W_{F,d}(0)=0$.
So a highly nonlinear power permutation $x\mapsto x^d$ of the finite field $F$ is one where $|W_{F,d}(a)|$ is small for all $a \in F$.
We shall see later (in Lemma \ref{Jacob}) that it is not possible to have all $|W_{F,d}(a)|$ less than or equal to $\sqrt{\card{F}}$, but Table \ref{Nancy} in Section \ref{Gerald} exhibits cases in which we can make every $|W_{F,d}(a)|$ value less than or equal to $\sqrt{p \card{F}}$, where $p$ is the characteristic of $F$.

\section{Crosscorrelation of m-sequences and Gold sequences}\label{Martha}

Linear recursive sequences over finite fields have been employed to produce pseudorandom sequences as early as the 1950's by Gilbert, Golomb, Welch, and Zierler (see the Preface of \cite{Golomb} and the Historical Introduction of \cite{Golomb-Gong} for brief historical overviews).
Much of the mathematical theory of these sequences had been worked out somewhat earlier: see \cite{Carmichael,Engstrom,Ward-31,Ward-33,Hall}.
From this theory, one can show that a sequence of elements from the field $\F_p$ of prime order $p$ satisfying a linear recursion whose minimal polynomial is of degree $n$ over $\F_p$ has a period less than or equal to $p^n-1$, with equality if and only if the minimal polynomial is a primitive irreducible.
In this case, the sequence is called a {\it maximum length linear feedback shift register sequence}, or a {\it maximal linear sequence}, or an {\it m-sequence}.

Golomb had given criteria for evaluating the pseudorandomness of binary sequences \cite[p.~25]{Golomb}, which can be generalized to nonbinary sequences \cite[\S5.1.2]{Golomb-Gong}.
One of these criteria is that the sequence should have low autocorrelation at nontrivial shifts; this property makes these sequences useful for applications to remote sensing and communications.
When communications systems have many users, it also becomes useful to have families of such sequences that have the additional property of low crosscorrelation between any pair of sequences in the family.
The m-sequences have very low autocorrelation, and pairs of m-sequences with the same period can be found that have relatively low crosscorrelation, and these in turn can be used to construct larger families of sequences (Gold sequences) in which every pair of sequences has low periodic crosscorrelation.

A sequence with terms from the prime field $\F_p$ is called a {\it $p$-ary sequence}.
It is natural to index the terms of any $p$-ary m-sequence of period $p^n-1$ using ${\mathbb Z}/(p^n-1){\mathbb Z}$.
If we fix $\alpha$, a primitive element of order $p^n-1$ in the finite field $F$ of order $p^n$, and let $\Tr \colon F \to \F_p$ be the absolute trace, then any $p$-ary m-sequence of period $p^n-1$ can be written as
\begin{equation}\label{Albert}
(\Tr(\alpha^{d j+s}))_{j \in {\mathbb Z}/(p^n-1){\mathbb Z}}
\end{equation}
for some integers $s$ and $d$ with $\gcd(d,p^n-1)=1$.
Let us take the m-sequence
\begin{equation}\label{Rachel}
(\Tr(\alpha^j))_{j \in {\mathbb Z}/(p^n-1){\mathbb Z}}
\end{equation}
with $d=1$ and $s=0$ as our reference sequence.
Then for any $d$ with $\gcd(d,p^n-1)=1$, the m-sequence
\begin{equation}\label{Desdemona}
(\Tr(\alpha^{d j}))_{j \in {\mathbb Z}/(p^n-1){\mathbb Z}}
\end{equation}
is obtained from our reference sequence by taking every $d$th element (proceeding cyclically according to our indexing): we call this new sequence the {\it decimation by $d$} of the reference sequence.
And then we see that a generic m-sequence \eqref{Albert} of period $p^n-1$ is obtained from our reference sequence by decimating by $d$ and then shifting (cyclically) by $s$ places.
When $d$ is a power of $p$ modulo $p^n-1$, the decimated sequence is identical to the reference sequence, and we say that such a decimation is {\it trivial}.

We shall now study the correlation properties of m-sequences, after setting down the basic definitions.
\begin{definition}[Periodic Crosscorrelation]
Let $\ell$ be a positive integer, and let $f=(f_j)_{j \in {\mathbb Z}/\ell{\mathbb Z}}$ and $g=(g_j)_{j \in {\mathbb Z}/\ell{\mathbb Z}}$ be $p$-ary sequences whose periods are divisors of $\ell$.
For any $s \in {\mathbb Z}/\ell{\mathbb Z}$, the {\it periodic crosscorrelation of $f$ with $g$ at shift $s$}, denoted $C_{f,g}(s)$, is defined to be
\[
C_{f,g}(s)=\sum_{j \in {\mathbb Z}/\ell{\mathbb Z}} e^{2\pi i (f_{j+s}-g_j)/p}.
\]
\end{definition}
We are interested in the crosscorrelation values at all the various shifts.
\begin{definition}[Crosscorrelation Spectrum]
Let $\ell$ be a positive integer, and let $f=(f_j)_{j \in {\mathbb Z}/\ell{\mathbb Z}}$ and $g=(g_j)_{j \in {\mathbb Z}/\ell{\mathbb Z}}$ be $p$-ary sequences whose periods are divisors of $\ell$.
The {\it crosscorrelation spectrum} of the sequence pair $(f,g)$ is $\{C_{f,g}(s): s \in {\mathbb Z}/\ell{\mathbb Z}\}$.
\end{definition}
It turns out that crosscorrelation values are character sums when $f$ and $g$ are m-sequences.
Let $g$ be our reference m-sequence \eqref{Rachel} and $f$ be \eqref{Desdemona} obtained by decimating $g$ by $d$.
We let $\psi\colon F \to {\mathbb C}$ be the canonical additive character, $\psi(x)=e^{2\pi i \Tr(x)/p}$, of our field $F$ of characteristic $p$ and order $p^n$.
Then
\begin{align}
C_{f,g}(s)
& =\sum_{j \in {\mathbb Z}/(p^n-1){\mathbb Z}} \psi(\alpha^{d (j+s)}-\alpha^j) \\
& =\sum_{j \in {\mathbb Z}/(p^n-1){\mathbb Z}} \psi(\alpha^{d j}-\alpha^{j-s}) \\
& =\sum_{x \in F^*}\psi(x^d-\alpha^{-s} x) \\
& =-1+W_{F,d}(\alpha^{-s}),
\end{align}
where $W_{F,d}$ is the Weil sum from Definition \ref{William}.
Thus the crosscorrelation spectrum for an m-sequence of period $p^n-1$ and its decimation by $d$ is given by the values of $-1+W_{F,d}(a)$ as $a$ runs through $F^*$, that is, the crosscorrelation spectrum is obtained from the Weil spectrum by subtracting $1$ from every value.
Note that when the decimation $d=1$, we are crosscorrelating our m-sequence with itself (this is called {\it autocorrelation}).
In this case the autocorrelation values $-1+W_{F,1}(a)$ all equal $-1$, except when $a=1$ (which corresponds to zero shift, and has autocorrelation $\card{F}-1=p^n-1$).
The property that autocorrelation at every nontrivial shift has a value of $-1$ is called the {\it ideal two-level autocorrelation} property of m-sequences, and is useful in applications involving synchronization.

If we continue to identify $g$ and $f$ with the m-sequences from \eqref{Rachel} and \eqref{Desdemona}, then for each $b \in F$, we can define the sequence
\[
h^{(b)}= (\Tr(\alpha^{d j}-b\alpha^j))_{j \in {\mathbb Z}/(p^n-1){\mathbb Z}},
\]
and we introduce the symbol $\infty$, and define
\[
h^{(\infty)} = g = (\Tr(\alpha^j))_{j \in {\mathbb Z}/(p^n-1){\mathbb Z}}.
\]
This gives a family $\{h^{(b)}: b \in F\cup\{\infty\}\}$ of $p^n+1$ sequences of period $p^n-1$ known as a family of {\it Gold sequences} \cite{Gold-67}.\footnote{Some authors only refer to the family so produced as a family of Gold sequences if the decimation $d$ is of a particular form originally selected by Gold, which produces favorable correlation spectra; but in the final portion of Gold's paper \cite{Gold-67}, the construction is presented in a way that a reader could straightforwardly generalize to an arbitrary decimation.}
For any $b \in F\cup\{\infty\}$, we can compute the autocorrelation function of the Gold sequence $h^{(b)}$ to obtain
\[
C_{h^{(b)},h^{(b)}}(s) = \begin{cases}
p^n-1 & \text{if $s=0$,} \\
-1 & \text{if $b=\infty$ and $s\not=0$,} \\
-1+W_{F,d}\left(\frac{b(\alpha^s-1)}{(\alpha^{d s}-1)^{1/d}}\right) & \text{if $b\not=\infty$ and $s\not=0$,}
\end{cases}
\]
where $1/d$ indicates the multiplicative inverse of $d$ modulo $\card{F^*}$.
So the autocorrelation values at nonzero shifts that one obtains among the various Gold sequences in our family are given by values of $-1+W_{F,d}(a)$ as $a$ runs through $F$ (recall that $W_{F,d}(0)=0$ while considering the penultimate case).
Now we suppose that $b$ and $c$ are two distinct elements of $F \cup\{\infty\}$, and compute the crosscorrelation between $h^{(b)}$ and $h^{(c)}$ at shift $s \in {\mathbb Z}/(p^n-1){\mathbb Z}$ to be
\[
C_{h^{(b)},h^{(c)}}(s) = \begin{cases}
-1+W_{F,d}(b+\alpha^{-s}) & \text{if $b\not=\infty$ and $c=\infty$,} \\
-1+W_{F,d}(c+\alpha^s) & \text{if $b=\infty$ and $c\not=\infty$,} \\
-1 & \text{if $b,c\not=\infty$, and $s=0$,} \\
-1+W_{F,d}\left(\frac{b\alpha^s-c}{(\alpha^{d s}-1)^{1/d}}\right) & \text{if $b,c\not=\infty$ and $s\not=0$,}
\end{cases}
\]
so we see that the crosscorrelation values one obtains among pairs of distinct Gold sequences in our family are given by values of $-1+W_{F,d}(a)$ as $a$ runs through $F$ (and again, recall that $W_{F,d}(0)=0$ while considering the penultimate case).

In applications involving Gold sequences, we want the magnitudes of autocorrelations at nonzero shifts and of crosscorrelations at all shifts to be small.
This is equivalent to saying that we want $|-1+W_{F,d}(a)|$ to be small for all $a \in F$, which is essentially the same goal we stated at the end of Section \ref{Persephone} for obtaining highly nonlinear power permutations (which was that $|W_{F,d}(a)|$ be small for all $a \in F$).

\section{Other applications}\label{Elijah}

Suppose that $F$ is the finite field of characteristic $p$ and order $q=p^n$ with primitive element $\alpha$, and let $d$ be a positive integer with $\gcd(d,q-1)=1$.
If $d \equiv 1 \pmod{p-1}$ and $d$ is not congruent to a power of $p$ modulo $q-1$, then the values of the Weil sum $W_{F,d}(a)$ in Definition \ref{William} as $a$ runs through $F$ determine the weight distribution of the $p$-ary cyclic error-correcting code ${\mathcal C}_{1,d}$ of length $p^n-1$ and dimension $2 n$ whose check polynomial is the product of the minimal polynomials of $\alpha^{-1}$ and $\alpha^{-d}$ over $\F_p$ (see \cite[\S A.3]{Katz-12}).
This code is the sum of two simplex codes, ${\mathcal C}_1$ and ${\mathcal C}_d$, whose respective check polynomials are the minimal polynomials of $\alpha^{-1}$ and $\alpha^{-d}$ over $\F_p$.
The nonzero words of these simplex codes all have weights of $(p-1) q/p$, while the words of ${\mathcal C}_{1,d}\smallsetminus ({\mathcal C}_1 \cup {\mathcal C}_d)$ have weights of the form
\begin{equation}\label{Ulrich}
\frac{(p-1) (q-W_{F,d}(a))}{p},
\end{equation}
where $a$ runs through $F^*$.
Since $W_{F,d}(0)=0$, we can summarize the above facts by saying that the nonzero words of ${\mathcal C}_{1,d}$ have weights given by \eqref{Ulrich} as $a$ runs through $F$.

When $p=2$, the values of the Weil sum $W_{F,d}(a)$ in Definition \ref{William} also have an application in finite geometry (see \cite[\S A.4]{Katz-12}).
They give the cardinalities of the intersections of hyperplanes with certain constructions described by Games in \cite{Games-m-sequences,Games-quadrics}.

\section{Galois Action of the Finite Field}\label{Bartholomew}

In this section we shall give known results about the Weil sum $W_{F,d}(a)$ from Definition \ref{William} arising from the Galois theory of the finite field $F$.
\begin{lemma}
If $\sigma$ is any automorphism of the finite field $F$ and $a \in F$, then $W_{F,d}(\sigma(a))=W_{F,d}(a)$.  In other words, if $F$ is of characteristic $p$, then $W_{F,d}(a^{p^k})=W_{F,d}(a)$ for every $a \in F$ and every integer $k$.
\end{lemma}
\begin{lemma}\label{Fiona}
If $F$ is of characteristic $p$ and order $q$, and $e \equiv p^k d \pmod{q-1}$ for some integer $k$, then $W_{F,e}(a)=W_{F,d}(a)$ for every $a \in F$.
\end{lemma}
See \cite[Lemmata 3.1--3.2]{Aubry-Katz-Langevin-15} for proofs of these two lemmata: the equivalent results in terms of crosscorrelation of m-sequences are well-known and go back at least as far as \cite[Theorem 3.1(d)--(e)]{Helleseth-76}.
Lemma \ref{Fiona} shows that if $d$ is a power of $p$ modulo $q-1$, then $W_{F,d}(a)=W_{F,1}(a)=\sum_{x \in F} \psi((1-a) x)$, so that we have a Weil sum of a monomial, which prompts the following definition.
\begin{definition}
If $F$ is a finite field of characteristic $p$ and order $q$, a positive integer $d$ congruent to a power of $p$ modulo $q-1$ is said to be a {\it degenerate exponent over $F$}.
\end{definition}
It is straightforward to determine which fields support nondegenerate invertible exponents.
\begin{lemma}\label{Hubert}
Let $F$ be a finite field.
If $\card{F} \leq 4$, every invertible exponent over $F$ is degenerate.
If $\card{F} > 4$, then there is a nondegenerate invertible exponent over $F$, for example, $\card{F}-2$.
\end{lemma}
The values of a degenerate Weil sum are very straightforward to calculate.
\begin{corollary}\label{Deidre}
If $d$ is degenerate over $F$, then
\[
W_{F,d}(a) = \begin{cases}
\card{F} & \text{if $a=1$,} \\
0 & \text{otherwise.}
\end{cases}
\]
\end{corollary}
The equivalent result in terms of crosscorrelation is \cite[Theorem 3.1(g)]{Helleseth-76}.
Using reparameterizations akin to those discussed in Section \ref{Wally}, we have the following result.
\begin{lemma}
If $d$ is invertible over a finite field $F$, and if $1/d$ denotes the inverse of $d$ modulo $\card{F^*}$, then $W_{F,1/d}(a)=W_{F,d}(a^{1/d})$ for every $a \in F$.
\end{lemma}
This means that $\{W_{F,1/d}(a): a \in S\}=\{W_{F,d}(a): a \in S\}$ for both $S=F$ and $S=F^*$.
Lemma \ref{Fiona} also gives cases where two exponents produce equivalent Weil sums, so we formulate the following definition.
\begin{definition}\label{Elizabeth}
If $d$ and $e$ are invertible exponents over a finite field $F$ of characteristic $p$ such that either $d \equiv p^k e \pmod{\card{F^*}}$ or $d \equiv p^k e^{-1} \pmod{\card{F^*}}$ for some integer $k$, then $d$ and $e$ are said to be {\it equivalent exponents over $F$}.
\end{definition}
The following result is the compositum of Lemma 3.3 and Corollary 3.4 of \cite{Aubry-Katz-Langevin-15}; note that we make some sign changes here to reflect the different sign convention we use in the definition of our Weil sum of a binomial.
\begin{lemma}
Let $F$ be a finite field of characteristic $p$, and let $E$ be an extension of $F$ with $[E:F]$ a power of a prime $\ell$ different from $p$.  If $1/d$ denotes the multiplicative inverse of $d$ modulo $p-1$, then
\[
W_{E,d}(a) \equiv W_{F,d}([E:F]^{1-1/d} a) \pmod{\ell}
\]
for every $a \in F$.  In particular, if $d$ is degenerate over $F$, then
\[
W_{E,d}(a) \equiv \begin{cases}
\card{F} \pmod{\ell} & \text{if $a=1$} \\
0 \pmod{\ell} & \text{if $a \in F\smallsetminus\{1\}$.}
\end{cases}
\]
\end{lemma}

\section{Galois Action of the Cyclotomic Field}\label{Cecilia}

The canonical additive character of a finite field $F$ of characteristic $p$ maps $F$ into the group of $p$th roots of unity in ${\mathbb C}$.
We set $\zeta_p = e^{2\pi i/p}$, so that our Weil sum $W_{F,d}(a)$ from Definition \ref{William} lies in ${\mathbb Z}[\zeta_p]$.
This is the ring of algebraic integers in the cyclotomic field ${\mathbb Q}(\zeta_p)$.
Thus we can take advantage of the Galois theory of this cyclotomic field to obtain results about our Weil sum.
First we note that an automorphism of the cyclotomic field permutes the spectrum of Weil sum values.
\begin{lemma}
Let $F$ be a finite field of characteristic $p$, and let $d$ be an invertible exponent over $F$.  If $\sigma \in \Gal({\mathbb Q}(\zeta_p)/{\mathbb Q})$ with $\sigma(\zeta_p)=\zeta_p^j$, then $\sigma(W_{F,d}(a))=W_{F,d}(j^{1-1/d} a)$ for every $a \in F$, where $1/d$ denotes the multiplicative inverse of $d$ modulo $p-1$.
\end{lemma}
The above lemma is proved in \cite[Theorem 2.1(b)]{Katz-12}, and it has the following corollaries (see \cite[Theorem 3.1(a)]{Helleseth-76} and \cite[Corollary 5.2]{Aubry-Katz-Langevin-15}, respectively).
\begin{corollary}
If $F$ is a finite field and $d$ is an invertible exponent over $F$, then $W_{F,d}(a)$ is a real number for every $a \in F$.
\end{corollary}
\begin{corollary}
Let $F$ be a finite field of characteristic $p$, and let $d$ be an invertible exponent over $F$.  If $A, B \in {\mathbb Q}(\zeta_p)$ are Galois conjugates over ${\mathbb Q}$, then the number of $a \in F^*$ such that $W_{F,d}(a)=A$ is equal to the number of $a \in F^*$ such that $W_{F,d}(a)=B$.
\end{corollary}
For certain $d$, the Weil sum values lie in a proper subfield of ${\mathbb Q}(\zeta_p)$.
\begin{lemma}
Let $F$ be a finite field of characteristic $p$, and let $d$ be an invertible exponent over $F$.
Let $K$ be the field generated by adjoining the Weil sum values $\{W_{F,d}(a): a \in F\}$ to ${\mathbb Q}$.
Then $[{\mathbb Q}(\zeta_p):K]$ is the largest divisor $m$ of $p-1$ such that $d \equiv 1 \pmod{m}$; since ${\mathbb Q}(\zeta_p)$ is a cyclic extension of ${\mathbb Q}$, this uniquely identifies $K$.
\end{lemma}
This was proved in \cite[Lemma 5.3]{Aubry-Katz-Langevin-15}, and has the following well known consequence.
\begin{corollary}
Let $F$ be a finite field of characteristic $p$, and let $d$ be an invertible exponent over $F$.
Then $\{W_{F,d}(a): a \in F^*\} \subseteq {\mathbb Z}$ if and only if $d \equiv 1 \pmod{p-1}$.
\end{corollary}
The version of this for crosscorrelation was proved as \cite[Theorem 4.2]{Helleseth-76}.

\section{Moments}\label{Daria}
Often it is difficult to understand the behavior of the individual values $W_{F,d}(a)$, but it is easier to compute and understand quantities determined by their collective behavior.
\begin{definition}[Power Moment]
If $k$ is a nonnegative integer, then the {\it $k$th power moment of $W_{F,d}$}, denoted $P_{F,d}^{(k)}$, is the sum of the $k$th powers of the Weil spectrum, that is, $P_{F,d}^{(k)}=\sum_{a \in F^*} W_{F,d}(a)^k$.
\end{definition}
A more general quantity was explored in the context of crosscorrelation in \cite[Theorem 3.2]{Helleseth-76}, and we state an equivalent result in terms of Weil sums here.
\begin{lemma}\label{Ursula}
Let $F$ be a finite field and $d$ an invertible exponent over $F$.
If $k$ is a positive integer and $b_1,\ldots,b_k \in F^*$, then
\[
\sum_{a \in F^*} \prod_{j=1}^k W_{F,d}(b_j a) = \frac{\card{F}^2 N_{b_1,\ldots,b_k}-\card{F}^k}{\card{F}-1},
\]
where $N_{b_1,\ldots,b_k}$ is the number of solutions $(x_1,\ldots,x_k) \in F^k$ to the system of equations
\begin{align*}
b_1 x_1 + \cdots + b_k x_k & = 0 \\
x_1^d+\cdots +x_k^d & =0.
\end{align*}
\end{lemma}
Note that $N_{b_1,\ldots,b_k}$ in the preceding lemma is the size of the intersection of a hyperplane and a Fermat variety in $k$-dimensional affine space.
When all the $b_j$'s are equal, we obtain the power moments.
\begin{corollary}
Let $F$ be a finite field and $d$ an invertible exponent over $F$.
If $k$ is a positive integer, then the $k$th power moment of $W_{F,d}$ is
\[
P_{F,d}^{(k)} = \sum_{a \in F^*} W_{F,d}(a)^k = \frac{\card{F}^2 N^{(k)} - \card{F}^k}{\card{F}-1},
\]
where $N^{(k)}$ is the number of solutions $(x_1,\ldots,x_k) \in F^k$ to the system of equations
\begin{align*}
x_1 + \cdots +x_k & = 0 \\
x_1^d+\cdots +x_k^d & =0.
\end{align*}
\end{corollary}
The first few power moments are of particular interest; see for example \cite[Corollary 2.6]{Katz-15}.
\begin{corollary}\label{Paul}
Let $F$ be a finite field and $d$ an invertible exponent over $F$.
\begin{enumerate}[(i)]
\item $P_{F,d}^{(0)} = \sum_{a \in F^*} W_{F,d}(a)^0 = \card{F}-1$,
\item\label{Frederick} $P_{F,d}^{(1)} = \sum_{a \in F^*} W_{F,d}(a)^1 = \card{F}$,
\item\label{Stephen} $P_{F,d}^{(2)} = \sum_{a \in F^*} W_{F,d}(a)^2 = \card{F}^2$,
\item $P_{F,d}^{(3)} = \sum_{a \in F^*} W_{F,d}(a)^3 = \card{F}^2 M_1$, and
\item $P_{F,d}^{(4)} = \sum_{a \in F^*} W_{F,d}(a)^4 = \card{F}^2 \sum_{a \in F^*} M_a^2$,
\end{enumerate}
where
\[
M_a = \card{\{x \in F: x^d+(1-x)^d=a\}}.
\]
\end{corollary}
The count $M_a$ in the last corollary is equal to the number of $x \in F$ such that
\[
(x+1)^d-x^d=a,
\]
which one sees by reparameterizing with $-x$ in place of $x$ (and noting that $\gcd(d,\card{F}-1)=1$ makes $d$ odd when $F$ is of odd characteristic).
Thus the power moments of $W_{F,d}$ are intimately connected to the differential spectrum of the power permutation $x\mapsto x^d$ of $F$.
Analysis of the differential spectrum has proved a powerful tool in studying $W_{F,d}$.
For example, Dobbertin's work in \cite{Dobbertin-99-Welch,Dobbertin-99-Niho} was a crucial step in verifying that certain $F$ and $d$ produce very favorable Walsh spectra, which are recorded later in this paper as the seventh and ninth entries of Table \ref{Nancy} in Section \ref{Gerald}.

\section{Archimedean Bounds}\label{Edwin}

In the Introduction we stated that in applications it is desirable to have an invertible exponent $d$ over a finite field $F$ such that $|W_{F,d}(a)|$ is small for all values of $a \in F^*$.
(Since $W_{F,d}(0)=0$, this is the same as saying that $W_{F,d}(a)$ is small for all $a \in F$.)
Since $W_{F,d}(a)$ is a sum of $\card{F}$ roots of unity in ${\mathbb C}$, it is clear that we always have $|W_{F,d}(a)|\leq \card{F}$.
In fact, the only way to achieve equality is if $d$ is degenerate (see \cite[Theorem 2.1(f)]{Katz-12} and Corollary \ref{Deidre} above).
\begin{lemma}\label{Martin}
Let $F$ be a finite field and $d$ an invertible exponent over $F$.
Then $|W_{F,d}(a)|=\card{F}$ if and only if $d$ is degenerate and $a=1$; otherwise $|W_{F,d}(a)| < \card{F}$.
\end{lemma}
One interesting consequence of this fact, in combination with the values of the first and second power moments in Corollary \ref{Paul}\eqref{Frederick}--\eqref{Stephen}, is that a nondegenerate $W_{F,d}$ must assume at least one positive and at least one negative value \cite[Corollary 2.3]{Aubry-Katz-Langevin-14}.
\begin{lemma}\label{Julie}
If $F$ is a finite field and $d$ is a nondegenerate invertible exponent over $F$, then $W_{F,d}(a) > 0$ for some $a \in F$ and $W_{F,d}(b) < 0$ for some $b$ in $F$.
\end{lemma}
Corollary \ref{Paul}\eqref{Stephen} also implies a useful standard about how small we can hope to make all the Weil sum values.
\begin{lemma}\label{Jacob}
For any finite field $F$ and invertible exponent $d$ over $F$, there is some $a \in F$ such that $|W_{F,d}(a)| > \sqrt{\card{F}}$.
\end{lemma}
This result, combined with the fact that there are known $F$ and $d$ such that $|W_{F,d}(a)| \leq \sqrt{2 \card{F}}$ for all $a \in F$ (for examples, see Table \ref{Nancy} in Section \ref{Gerald}), means that ``small'' should understood as ``not much larger than $\sqrt{\card{F}}$.''

The Weil-Carlitz-Uchiyama bound on character sums \cite{Weil,Carlitz-Uchiyama} can be applied to our Weil sum.
\begin{theorem}
Let $F$ be a finite field and $d$ a nondegenerate invertible exponent over $F$.
Then $|W_{F,d}(a)|\leq (d-1)\sqrt{\card{F}}$ for every $a \in F$.
\end{theorem}
Of course, when $d \geq 1+\sqrt{\card{F}}$, this bound is trivial.
When $d=\card{F}-2$, we see that
\[
W_{F,\card{F}-2}(a)=1+\sum_{x \in F^*} \psi(x^{-1}-a x),
\]
that is, one plus a Kloosterman sum.  The results of Weil-Carlitz-Uchiyama \cite{Weil,Carlitz-Uchiyama} can also be applied in this case.
\begin{theorem}
If $F$ is a finite field, then $|W_{F,\card{F}-2}(a)|\leq 2\sqrt{\card{F}}$ for every $a \in F$.
\end{theorem}
Interestingly, if $F$ is of characteristic $2$, then Lachaud and Wolfmann \cite{Lachaud-Wolfmann} obtained the precise spectrum.
\begin{theorem}\label{Otto}
If $F$ is a finite field of characteristic $2$ and order $q$, then $\{W_{F,\card{F}-2}(a): a \in F^*\}$ is the set $[1-2\sqrt{q},1+2\sqrt{q}]\cap (4 {\mathbb Z})$.
\end{theorem}
And Katz and Livn\'e \cite{Katz-Livne} obtained the analogous result in characteristic $3$.
\begin{theorem}
If $F$ is a finite field of characteristic $3$ and order $q$, then $\{W_{F,\card{F}-2}(a): a \in F^*\}$ is the set $[1-2\sqrt{q},1+2\sqrt{q}]\cap (3 {\mathbb Z})$.
\end{theorem}
If the characteristic of $F$ is greater than $3$, Kononen, Rinta-aho, and V\"a\"an\"anen \cite{Kononen-Rinta-aho-Vaananen} showed a markedly different behavior.
\begin{theorem}
If $F$ is a finite field of characteristic $p > 3$, then there is no $a \in F^*$ such that $W_{F,\card{F}-2}(a)=0$.
\end{theorem}
In characteristic $2$, a stronger version of the lower bound in Lemma \ref{Jacob} was deduced by Pursley and Sarwate \cite[eqs.~(4.6)--(4.8)]{Sarwate-Pursley} by applying a bound of of Sidel{\cprime}nikov \cite{Sidelnikov} to the family of Gold sequences (see Section \ref{Martha}) whose correlation spectra derive from $W_{F,d}$.
\begin{theorem}\label{Ignatius}
Let $F$ be a finite field of characteristic $2$ and $d$ an invertible exponent over $F$.
Then there is some $a \in F^*$ such that $|W_{F,d}(a)-1| > \sqrt{2(\card{F}-2)}$.
If $[F:{\mathbb F}_2]$ is odd and greater than one, this means that there is some $a \in F^*$ such that $|W_{F,d}(a)-1| \geq 1+\sqrt{2 \card{F}}$.
\end{theorem}
The $-1$ terms that appear in these bounds are a consequence of the fact that the correlation values are obtained from Weil sum values by subtracting $1$.
From these results about correlation, the following bound on the Weil sum itself immediately follows.
\begin{corollary}
Let $F$ be a finite field of characteristic $2$ with $[F:{\mathbb F}_2]$ odd, and let $d$ be an invertible exponent over $F$.
Then there is some $a \in F^*$ such that $|W_{F,d}(a)| \geq \sqrt{2 \card{F}}$.
\end{corollary}
For every finite field $F$ of characteristic $2$ with $[F:{\mathbb F}_2]$ odd, there is some $d$ such that $W_{F,d}(a) \in \{0,\pm\sqrt{2 \card{F}}\}$ for all $a \in F$ (see the first, third, seventh, and ninth entries of Table \ref{Nancy} in Section \ref{Gerald} for examples), so the above corollary gives a sharp bound in this case.
When $[F:{\mathbb F}_2]$ is even, then Sarwate and Pursley \cite[\S3]{Sarwate-Pursley} suspect that the bound of Theorem \ref{Ignatius} is weak, and conjecture that it is not possible to keep all the values of $|W_{F,d}(a)-1|$ smaller than what one observes when $d=\card{F}-2$ in Theorem \ref{Otto}.
\begin{conjecture}[Sarwate-Pursley, 1980]
Let $F$ be of characteristic $2$ with $[F:{\mathbb F}_2]$ even, and let $d$ be an invertible exponent over $F$.
Then there is some $a \in F^*$ such that $|W_{F,d}(a)-1| \geq -1 + 2\sqrt{\card{F}}$, or equivalently, there is some $a \in F^*$ such that $|W_{F,d}(a)| \geq 2\sqrt{\card{F}}$.
\end{conjecture}
The equivalent version, not explicitly stated by Sarwate and Pursley, arises because every value $W_{F,d}(a)$ is a multiple of $4$ when $F$ is a non-prime field of characteristic $2$ (see Corollary \ref{Eugene}).
Feng, Leung, and Xiang \cite{Feng-Leung-Xiang} proved the following interesting result.
\begin{theorem}\label{Helen}
If $F$ is of characteristic $2$ and order $2^{2 m}$, then there is some $a \in F^*$ such that $W_{F,d}(a) > 2^m+2^{\floor{m/2}}$.
\end{theorem}
In terms of the magnitudes $|W_{F,d}(a)|$, this is stronger than Theorem \ref{Ignatius} when $\card{F} \leq 64$.
But interestingly, the bound on $W_{F,d}(a)$ in Theorem \ref{Helen} does not involve an absolute value, so it tells us something specific about positive values of $W_{F,d}$.

\section{Non-Archimedean Bounds}\label{Felix}

The values $W_{F,d}(a)$ of our Weil sums lie in the cyclotomic field ${\mathbb Q}(\zeta_p)$, where $p$ is the characteristic of $F$ and $\zeta_p$ is a primitive $p$th root of unity.
The previous section dealt with Archimedean bounds, that is, bounds on the magnitude of our Weil sums with respect to the absolute value from the larger field ${\mathbb C}$.
We can also view our Weil sums $p$-adically, and ask for non-Archimedean bounds, that is, bounds on $p$-divisibility.
To this end, we recall that the {\it $p$-adic valuation} of a nonzero $a \in {\mathbb Z}$ is the unique $k$ such that $p^k \mid a$ and $p^{k+1}\nmid a$, where one defines $v_p(0)=\infty$, and then extends $v_p$ to ${\mathbb Q}$ by declaring that $v_p(a/b)=v_p(a)-v_p(b)$ for every $a,b \in {\mathbb Z}$ with $b\not=0$.
One should recall from algebraic number theory that ${\mathbb Z}[\zeta_p]$ is the ring of algebraic integers of the field ${\mathbb Q}(\zeta_p)$.
In ${\mathbb Z}[\zeta_p]$, the ideal $(p)$ is the $(p-1)$th power of the prime ideal $(1-\zeta_p)$.
This allows us to extend the usual $p$-adic valuation $v_p$ from ${\mathbb Q}$ to ${\mathbb Q}(\zeta_p)$, so that $v_p(1-\zeta_p)=1/(p-1)$.
We are interested in the $p$-adic valuations of Weil sum values $W_{F,d}(a)$.
\begin{definition}[Valuation of Weil Spectrum $V_{F,d}$]
Let $F$ be a finite field and let $d$ be invertible over $F$.
Then the {\it valuation of the Weil spectrum of $F$ and $d$}, denoted $V_{F,d}$, is
\[
V_{F,d} = \min_{a \in F^*} v_p(W_{F,d}(a)).
\]
\end{definition}

To investigate $V_{F,d}$, it is useful to consider the Fourier expansion of the Weil sum in terms of Gauss sums.
If $F$ is a finite field of order $q$, we let $\zeta_{q-1}$ be a primitive $(q-1)$th root of unity over ${\mathbb Q}$, and then a {\it multiplicative character} is a group homomorphism $\chi\colon F^* \to {\mathbb Q}(\zeta_{q-1})^*$.
We let $\widehat{\Fu}$ denote the group of multiplicative characters of $F^*$, and $\id$ will always denote the trivial multiplicative character.
For any multiplicative character $\chi$ of a field $F$, the Gauss sum $G(\chi)$ is defined to be
\[
G(\chi)=\sum_{a \in F^*} \psi(a)\chi(a),
\]
where $\psi$ is the canonical additive character of $F$.
Thus $G(\chi) \in {\mathbb Q}(\zeta_p,\zeta_{q-1})$.
One can expand the Weil sum in terms of pairwise products of these Gauss sums, and vice-versa.
\begin{lemma}
Let $F$ be a finite field of order $q$, let $d$ be an invertible exponent over $F$, and use $\id$ to denote the trivial multiplicative character of $F$.
Then
\[
W_{F,d}(a) = \frac{q}{q-1} + \frac{1}{q-1} \sum_{\chi\not=\id} G(\chi) G(\conj{\chi}^d) \chi^d(-a),
\]
and
\[
\sum_{a \in F^*} W_{F,d}(a) \conj{\chi^d(-a)} = \begin{cases}
q & \text{if $\chi=\id$}, \\
G(\chi) G(\conj{\chi}^d) & \text{otherwise.}
\end{cases}
\]
\end{lemma}
For a proof, see \cite[\S4]{Aubry-Katz-Langevin-15}, where the differences between the expressions there and here are due to a sign difference in the definition of $W_{F,d}(a)$.\footnote{And for the same reason, the formulae in \cite[Lemma 2.5]{Katz-Langevin-Lee-Sapozhnikov} should have the same sign changes, but one can see that this does not affect the proof of Corollary 2.6 of that paper, which is the only place this result is used.}
A consequence of our expansions is that we can understand the valuation of the Weil spectrum in terms of the valuations of pairwise products of Gauss sums.
\begin{corollary}\label{Clarence}
Let $F$ be a finite field of characteristic $p$, and let $d$ be an invertible exponent over $F$.  If $\card{F}=2$, then $d$ is degenerate over $F$ and $V_{F,d}=1$.  If $\card{F} > 2$, then
\[
V_{F,d}=\mins{\chi\in\widehat{\Fu} \\ \chi\not=\id} v_p(G(\chi) G(\conj{\chi}^d).
\]
\end{corollary}
See \cite[Corollary 2.6]{Katz-Langevin-Lee-Sapozhnikov} for a proof.\footnote{Also see \cite[Lemma 4.1]{Aubry-Katz-Langevin-15}, which neglects the fact that the trivial case when $\card{F}=2$ must be handled separately; the rest of that paper is unaffected by this oversight because the lemma is used only in Corollary 4.2 of that paper, which remains true because $V_{L,d} \leq [L:{\mathbb F}_2]$ can be deduced immediately from the fact that the first power moment for $W_{L,d}$ (given in Lemma 2.1(i) of that paper) has $2$-adic valuation $[L:{\mathbb F}_2]$.}
A further corollary of this result can be obtained using the Davenport-Hasse relation \cite[Corollary 4.2]{Aubry-Katz-Langevin-15}.
\begin{corollary}
Let $K$ be a finite field, $F$ a finite extension of $K$, and $d$ an invertible exponent over $F$.  Then $V_{F,d} \leq [F:K] V_{K,d}$.
\end{corollary}

Corollary \ref{Clarence} tells us how to obtain the valuation of the Weil spectrum from those of pairwise products of Gauss sums.
Now Stickelberger's Theorem tells us the $p$-adic valuation of these Gauss sums, which in turn gives the exact value of $V_{F,d}$ in terms of a combinatorial problem concerning quantities known as $p$-ary weights.
If $p$ is a prime and $n$ is an integer, we define the {\it $p$-ary weight function on ${\mathbb Z}/(p^n-1){\mathbb Z}$}, to be the function $\wt \colon {\mathbb Z}/(p^n-1){\mathbb Z} \to {\mathbb Z}$ with
\[
\wt(a_0 p^0 +a_1 p^1 + \cdots + a_{n-1} p^{n-1}) = a_0 + a_1 + \cdots + a_{n-1}
\]
where the powers of $p$ in this expression are elements of ${\mathbb Z}/(p^n-1){\mathbb Z}$, while the coefficients $a_0,a_1,\ldots,a_{n-1}$ are elements of the subset $\{0,1,\ldots,p-1\}$ of ${\mathbb Z}$, with at least one $a_j < p-1$.
Since each element of ${\mathbb Z}/(p^n-1){\mathbb Z}$ has a unique expression of this form, this is a well-defined function.
Now we may state what Stickelberger's Theorem tells us about the value of $V_{F,d}$.
\begin{lemma}
Let $F$ be a finite field of characteristic $p$ and order $q=p^n > 2$, let $d$ be an invertible exponent over $F$, and let $\wt$ be the $p$-ary weight function for ${\mathbb Z}/(p^n-1){\mathbb Z}$.  Define
\[
m = \mins{j \in {\mathbb Z}/(q-1){\mathbb Z} \\ j\not=0} \big(\wt(j)+\wt(-d j)\big),
\]
or equivalently
\[
m = (p-1) n + \mins{j \in {\mathbb Z}/(q-1){\mathbb Z} \\ j\not=0} \big(\wt(d j)-\wt(j)\big).
\]
Then $V_{F,d} = m/(p-1)$.
\end{lemma}
When $d\equiv 1 \pmod{p-1}$ (which is invariably true in characteristic $2$), one can obtain an equivalent result from McEliece's Theorem \cite{McEliece-71,McEliece-72} on $p$-divisibility of weights in cyclic codes; earlier uses of this principle in characteristic $2$  often proceeded in this way \cite{McGuire-Calderbank,Calderbank-McGuire-Poonen-Rubinstein,Canteaut-Charpin-Dobbertin-99,Canteaut-Charpin-Dobbertin-00-Binary,Canteaut-Charpin-Dobbertin-00-Weight,Hollmann-Xiang,Hou,Dobbertin-Helleseth-Kumar-Martinsen}.  (But also see \cite{Langevin-Veron,Leander-Langevin} for approaches proceeding directly from Stickelberger's theorem in characteristic $2$.)
The more general result in the lemma here is stated and proved as \cite[Lemma 2.9]{Katz-Langevin-Lee-Sapozhnikov}.
The same result also appears as \cite[Proposition 4.3]{Katz-Langevin-15}, with the $\card{F} >2$ condition missing, which does not affect the other results of that paper.
One consequence of this result is that all values of our Weil sum have nontrivial $p$-divisibility, and we have some sense of the range of possible values for $V_{F,d}$.
\begin{corollary}\label{Eugene}
Let $F$ be a finite field of characteristic $p$ and let $d$ be an invertible exponent over $F$.
Then $V_{F,d}=[F:\F_p]$ if and only if $d$ is degenerate over $F$.
If $d$ is nondegenerate over $F$, then $2/(p-1) \leq V_{F,d} < [F:\F_p]$ with $V_{F,d}=2/(p-1)$ if and only if $d$ is an exponent equivalent to $\card{F}-2$ over $F$.
\end{corollary}
This was proved as \cite[Corollary 2.10]{Katz-Langevin-Lee-Sapozhnikov}.
Some stronger upper bounds for $V_{F,d}$ were also proved in \cite[Theorem 1.1, Remark 1.3]{Katz-Langevin-Lee-Sapozhnikov}.
\begin{theorem}\label{Julius}
Let $F$ be a finite field of characteristic $p$, and let $d$ be an invertible exponent over $F$.
\begin{enumerate}[(i)]
\item\label{Hans} If $d$ is degenerate over $F$, then $V_{F,d}=[F:\F_p]$.
\item\label{Edith} If $d$ is nondegenerate over $F$, but degenerate over $\F_p$, then we have the following:
\begin{enumerate}[(a)]
\item If\label{Ferdinand} $[F:\F_p]$ is a power of $2$, then $V_{F,d} \leq \frac{1}{2}[F:\F_p]$.
\item If\label{George} $[F:\F_p]$ is not a power of $2$, then $V_{F,d} \leq \frac{2}{3}[F:\F_p]$.
\end{enumerate}
\item\label{Henry} If $d$ is nondegenerate over $\F_p$ (which implies $p \geq 5$), then we have the following:
\begin{enumerate}[(a)]
\item\label{Irene} If $p\equiv 1\pmod{4}$ and $[F:\F_p]$ is odd, then $V_{F,d} \leq \frac{1}{2}[F:\F_p]$; but
\item\label{James} If $p\equiv 3\pmod{4}$ or $[F:\F_p]$ is even, then $V_{F,d} \leq \frac{1}{p-1} \ceil{\frac{p-1}{3}}[F:\F_p]$.
\end{enumerate}
\end{enumerate}
\end{theorem}
In many cases, the upper bounds of Theorem \ref{Julius} are the best possible, as discussed in \cite[Remark 1.3]{Katz-Langevin-Lee-Sapozhnikov}.
If we are not concerned about the degeneracy of the exponent over the prime field, we get the following simplified version of Theorem \ref{Julius}.
\begin{corollary}\label{Katherine}
Let $F$ be a finite field of characteristic $p$ and let $d$ be a nondegenerate invertible exponent over $F$.
Then $V_{F,d} \leq \frac{2}{3} [F:\F_p]$.
If $[F:\F_p]$ is a power of $2$, then $V_{F,d} \leq \frac{1}{2} [F:\F_p]$.
\end{corollary}
The following result \cite[Lemmata 4.1 and 4.2]{Katz-Langevin-Lee-Sapozhnikov} shows that the above bound is best possible when $[F:\F_p]$ is a power of $2$ greater than $1$.
\begin{theorem}
Let $F$ be a finite field of characteristic $p$ with $[F:\F_p]=2^s$ for some $s > 0$, and suppose that $F\not={\mathbb F}_4$.
Then there is a nondegenerate invertible exponent $d$ over $F$ such that $V_{F,d}=\frac{1}{2}[F:\F_p]$.
\end{theorem}
Recall from Lemma \ref{Hubert} that the condition $F\not={\mathbb F}_4$ is necessary for $F$ to have a nondegenerate exponent.
The general bound in Corollary \ref{Katherine} is best possible when $[F:\F_p]$ is divisible by $3$, as seen in the following result \cite[Lemma 3.2]{Katz-Langevin-Lee-Sapozhnikov}.
\begin{theorem}
Suppose $F$ is a finite field of characteristic $p$ with $[F:\F_p]$ not a power of $2$, and let $\ell$ be the smallest odd prime divisor of $[F:\F_p]$.
Then there is a nondegenerate invertible exponent $d$ over $F$ such that $V_{F,d}=\frac{\ell+1}{2\ell} [F:\F_p]$.
\end{theorem}
Katz, Langevin, Lee, and Sapozhnikov \cite[Conjecture 6.1]{Katz-Langevin-Lee-Sapozhnikov} conjecture that the $V_{F,d}$ values seen in this proposition are the highest possible.
\begin{conjecture}[Katz-Langevin-Lee-Sapozhnikov, 2017]\label{Lawrence}
If $F$ is a finite field of characteristic $p$ with $[F:\F_p]$ not a power of $2$, if $\ell$ is the smallest odd prime divisor of $[F:\F_p]$, and if $d$ is a nondegenerate invertible exponent over $F$, then $V_{F,d}\leq \frac{\ell+1}{2\ell} [F:\F_p]$.
\end{conjecture}
Conjecture \ref{Lawrence} proposes an upper bound that is often stronger than the $\frac{2}{3}[F:\F_p]$ bound of Corollary \ref{Katherine}, viz.~when $[F:\F_p]$ is greater than $10$ and is neither a power of $2$ nor a multiple of $3$.
Computer checks \cite[\S6]{Katz-Langevin-Lee-Sapozhnikov} have shown that Conjecture \ref{Lawrence} is true whenever $\card{F} < 10^{13}$.

\section{Number of Values}\label{Gerald}

In this section we are concerned with the cardinality of the Weil spectrum $\{W_{F,d}(a): a \in F^*\}$.
\begin{definition}[Exactly, at Least, and at Most $k$-Valued $W_{F,d}$]
Let $F$ be a finite field and let $d$ be invertible over $F$.
We say that $W_{F,d}$ (or the Weil spectrum of $F$ and $d$) is {\it $k$-valued} (or sometimes {\it exactly $k$-valued} for emphasis) to mean that $\card{\{W_{F,d}(a): a \in F^*\}}=k$.
And we say that $W_{F,d}$ is {\it at least $k$-valued} to mean that $\card{\{W_{F,d}(a): a \in F^*\}} \geq k$, or {\it at most $k$-valued} to mean that $\card{\{W_{F,d}(a): a \in F^*\}}\leq k$.
\end{definition}
If $d$ is degenerate over $F$, then Corollary \ref{Deidre} shows that $W_{F,d}$ is at most $2$-valued: to be precise, $W_{F,d}$ is $1$-valued when $F={\mathbb F}_2$ and $2$-valued for any other field.
A classic result of Helleseth \cite[Theorem 4.1]{Helleseth-76} shows that $W_{F,d}$ is at least three-valued if $d$ is nondegenerate over $F$.
\begin{theorem}\label{Sally}
Let $F$ be a finite field and $d$ an invertible exponent over $F$.
Then $W_{F,d}$ is at least three-valued if and only if $d$ is nondegenerate.
\end{theorem}
There has been a great deal of interest in determining when $W_{F,d}$ is exactly three-valued.
Table \ref{Nancy} displays known pairs $(F,d)$ such that $W_{F,d}$ is three-valued; any known pair $(F,d)$ with a three-valued $W_{F,d}$ is either listed on the table or else corresponds to some entry $(F,d')$ on the table with $d'$ equivalent to $d$ over $F$.
Note also that we use the $2$-adic valuation $v_2$ in describing some of the required conditions on the table.
\begin{table}
\caption{Three-valued Weil sums of binomials}\label{Nancy}
\begin{center}
{\setlength{\extrarowheight}{3pt}\begin{tabular}{cccc}
order $q$ of $F$ & exponent $d$ & Weil spectrum & reference \\
\hline
 & $d=2^i+1$ & &  \\\noalign{\vskip-0.2pt}
$q=2^n$ & $0 < i < n$ & $0,\pm\sqrt{2^{\gcd(n,i)} q}$ & \cite{Kasami-66,Kasami-Lin-Peterson,Gold-68} \\\noalign{\vskip-0.2pt}
& $v_2(i) \geq v_2(n)$ & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
 & $d=(p^{2 i}+1)/2$ & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
\multirow{-2}{*}{$q=p^n$} & $0 < i < n$ & $0,\pm\sqrt{p^{\gcd(n,i)} q}$ & \multirow{-2}{*}{\cite{Trachtenberg} ($n$ odd)} \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
\multirow{-2}{*}{$p$ odd} & $v_2(i) \geq v_2(n)$ &  & \multirow{-2}{*}{\cite{Helleseth-71,Helleseth-76} ($n$ even)}\\\noalign{\vskip-0.2pt}
 & $d=2^{2 i}-2^i+1$ & &  \\\noalign{\vskip-0.2pt}
$q=2^n$ & $0 < i < n$ & $0,\pm\sqrt{2^{\gcd(n,i)} q}$ & \cite{Welch,Kasami-71} \\\noalign{\vskip-0.2pt}
& $v_2(i) \geq v_2(n)$ & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
 & $d=p^{2 i}-p^i+1$ & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
\multirow{-2}{*}{$q=p^n$} & $0 < i < n$ & $0,\pm\sqrt{p^{\gcd(n,i)} q}$ & \multirow{-2}{*}{\cite{Trachtenberg} ($n$ odd)} \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
\multirow{-2}{*}{$p$ odd} & $v_2(i) \geq v_2(n)$ &  & \multirow{-2}{*}{\cite{Helleseth-71,Helleseth-76} ($n$ even)}\\\noalign{\vskip-0.2pt}
$q=2^n$ & & & \\\noalign{\vskip-0.2pt}
$n>2$ & $d=2^{n/2}+2^{(n+2)/4}+1$ & $0,\pm 2\sqrt{q}$ & \cite{Cusick-Dobbertin} \\\noalign{\vskip-0.2pt}
$v_2(n)=1$ & & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$q=2^n$ & & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$n>2$ & $d=2^{(n+2)/2}+3$ & $0,\pm 2\sqrt{q}$ & \cite{Cusick-Dobbertin} \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$v_2(n)=1$ & & & \\\noalign{\vskip-0.2pt}
$q=2^n$ & & & \\\noalign{\vskip-0.2pt}
$n>1$ & $d=2^{(n-1)/2}+3$ & $0,\pm \sqrt{2 q}$ & \cite{Canteaut-Charpin-Dobbertin-99,Canteaut-Charpin-Dobbertin-00-Binary,Hollmann-Xiang} \\\noalign{\vskip-0.2pt}
$n$ odd & & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$q=3^n$ & & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$n>1$ & $d=2\cdot 3^{(n-1)/2}+1$ & $0,\pm \sqrt{3 q}$ & \cite{Dobbertin-Helleseth-Kumar-Martinsen} \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$n$ odd & & & \\\noalign{\vskip-0.2pt}
$q=2^n$ & $d=2^{2 i}+2^i-1$ & & \\\noalign{\vskip-0.2pt}
$n>1$ & $0 < i < n$ & $0,\pm \sqrt{2 q}$ & \cite{Hollmann-Xiang,Hou} \\\noalign{\vskip-0.2pt}
$n$ odd & $i \equiv -1/4 \pmod{n}$ & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$q=3^n$ & $d=2 \cdot 3^i+1$ & & \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$n>1$ & $0 < i <  n$ & $0,\pm \sqrt{3 q}$ & \cite{Katz-Langevin-15} \\\noalign{\vskip-0.2pt}
\rowcolor{tablecolor}[5.2pt]
$n$ odd & $i \equiv -1/4 \pmod{n}$  &  &
\end{tabular}}
\end{center}
\end{table}
Some features in Table \ref{Nancy} become immediately apparent.
First of all, it should be noted that for every entry, the three values of $W_{F,d}(a)$ lie in ${\mathbb Z}$.
That this must be true whenever $W_{F,d}$ is three-valued was proved in \cite[Theorem 1.7]{Katz-12}, and a simpler proof can be found in \cite[\S9]{Aubry-Katz-Langevin-15}.
\begin{theorem}\label{Agnes}
If $F$ is a finite field, $d$ is an invertible exponent over $F$, and $W_{F,d}$ is three-valued, then $d \equiv 1 \pmod{p-1}$ and $W_{F,d}(a) \in {\mathbb Z}$ for all $a \in F$.
\end{theorem}
Another feature of note in Table \ref{Nancy} is that $0$ always appears as one of the values of $W_{F,d}(a)$ with $a \in F^*$.
This was proved in \cite[Theorem 1.9]{Katz-12}
\begin{theorem}\label{Norbert}
If $F$ is a finite field, $d$ is an invertible exponent over $F$, and $W_{F,d}$ is three-valued, then $W_{F,d}(a)=0$ for some $a \in F^*$.
\end{theorem}
Another notable feature of the values of $W_{F,d}(a)$ appearing on the table is that the two nonzero values are opposites.
Lemma \ref{Julie} tells us that the two nonzero values must have opposite signs, but does not require them to have the same magnitude.
We say that a three-valued $W_{F,d}$ is {\it symmetric} if the two nonzero values in $\{W_{F,d}(a): a \in F^*\}$ are opposites of each other.
It is not known whether all three-valued $W_{F,d}$ are symmetric, but we may conjecture it.
\begin{conjecture}\label{Simon}
If $F$ is a finite field, $d$ is an invertible exponent over $F$, and $W_{F,d}$ is three-valued, then it is symmetric.
\end{conjecture}
A much earlier conjecture looks at which fields $F$ have at least one invertible exponent $d$ such that $W_{F,d}$ is three-valued.
The first two entries of Table \ref{Nancy} show that for each field $F$ of characteristic $p$ and order $q=p^n$, one can obtain an exponent $d$ such that $W_{F,d}$ is three-valued, provided that $n$ is not a power of $2$.
And we note that when $n$ is a power of $2$, none of the entries of the table provide an exponent $d$ that makes $W_{F,d}$ three-valued.
This led Helleseth to conjecture (see \cite{Helleseth-71}, \cite[Conjecture 5.2]{Helleseth-76}) that fields of order $p^n$ with $n$ a power of $2$ do not support three-valued sums.
\begin{conjecture}[Helleseth Three-Valued Conjecture, 1971]\label{Thomas}
If $F$ is a finite field of characteristic $p$ with $[F:\F_p]$ a power of $2$, then there does not exist any invertible exponent $d$ over $F$ such that $W_{F,d}$ is three-valued.
\end{conjecture}
This conjecture has been studied by many researchers \cite{Calderbank-McGuire-Poonen-Rubinstein,McGuire,Charpin,CakCak-Langevin,Feng,Katz-12,Aubry-Katz-Langevin-14,Aubry-Katz-Langevin-15,Katz-15,Katz-Langevin-16}, and was proved to be true in characteristics $2$ and $3$ (see \cite[Corollary 1.10]{Katz-12} and \cite[Theorem 1.7]{Katz-15}).
\begin{theorem}\label{Alexandra}
If $F$ is a finite field of characteristic $p\in\{2,3\}$ with $[F:\F_p]$ a power of $2$, then there does not exist any invertible exponent $d$ over $F$ such that $W_{F,d}$ is three-valued.
\end{theorem}

Conjecture \ref{Thomas} remains open for characteristics $p \geq 5$, but some partial results have been obtained.
The following theorem (announced in \cite[Th\'eor\`eme 3.1]{Aubry-Katz-Langevin-14} and proved in \cite[Theorem 1.4]{Aubry-Katz-Langevin-15}) exhibits an obstruction to obtaining symmetric three-valued Weil sums.
\begin{theorem}\label{Linda}
Let $F$ be a finite field and $K$ and $L$ subfields of $F$ with $[L:K]=2$, and suppose that $d$ is an invertible exponent over $F$ that is nondegenerate over $L$ but degenerate over $K$.  Then $W_{F,d}$ is not symmetric three-valued.
\end{theorem}
A corollary of this theorem is that if there is any counterexample to Conjecture \ref{Thomas}, then it cannot be symmetric.
\begin{corollary}
If $F$ is a finite field of characteristic $p$ with $[F:\F_p]$ a power of $2$, and $d$ is an invertible exponent over $F$, then $W_{F,d}$ is not symmetric three-valued.
\end{corollary}
Thus Conjecture \ref{Simon} implies Conjecture \ref{Thomas}.
The characteristic $2$ case of this corollary was proved in \cite[Theorem 3]{Calderbank-McGuire-Poonen-Rubinstein}, and the general case was proved in \cite[Corollary 1.5]{Aubry-Katz-Langevin-15}.
A stronger version of the characteristic $2$ case due to Feng \cite[Theorem 2]{Feng} (stating that zero cannot appear in the Weil spectrum) was a critical advance toward proving Conjecture \ref{Thomas} in characteristic $2$; once this was established, the result that zero must occur (Theorem \ref{Norbert}) finished the proof of Helleseth's Three-Valued conjecture in characteristic $2$.

A symmetric three-valued Weil sum has many constraints on its three values, which we summarize here (see \cite[Proposition 1.3]{Aubry-Katz-Langevin-15}).
\begin{lemma}\label{David}
If $F$ is a finite field of characteristic $p$ and order $q$, and $d$ is an invertible exponent over $F$ such that $W_{F,d}$ is symmetric three-valued, then the three values must be $0$ and $\pm p^k$ for some positive integer $k$ such that $\sqrt{q} < p^k < q$.
\end{lemma}
A symmetric three-valued Weil sum is said to be {\it preferred} if the magnitude of the nonzero values is as small as Lemma \ref{David} allows it to be.
Thus when $F$ is of characteristic $p$ and order $q$, a preferred three-valued Weil sum has values $0$ and $\pm\sqrt{p q}$ if $q$ is an odd power of $p$, and has values $0$ and $\pm p\sqrt{q}$ if $q$ is an even power of $p$.
We note that the last six entries in Table \ref{Nancy} always provide preferred three-valued $W_{F,d}$ and the first four entries provide preferred three-valued $W_{F,d}$ for fields $F$ when $[F:\F_p]$ is not divisible by $4$ nor equal to $1$ or $2$.
In fact, the first two entries of Table \ref{Nancy} show that if $F$ has $[F:\F_p]$ not divisible by $4$ nor equal to $1$ or $2$, then there is some invertible exponent $d$ over $F$ such that $W_{F,d}$ is preferred three-valued.
The absence of preferred three-valued Weil sums when $[F:\F_p]$ is divisible by $4$ is is a consequence of a more general fact (see \cite[Th\'eor\`eme 3.3]{Aubry-Katz-Langevin-14} and \cite[Theorem 1.7]{Aubry-Katz-Langevin-15}) concerning the $2$-adic valuation of $[F:\F_p]$.
\begin{theorem}
Let $F$ be a finite field of characteristic $p$ and order $q$ with $v_2([F:\F_p])=s$.
If $d$ is an invertible exponent over $F$ such that $W_{F,d}$ is symmetric three-valued, then the three values must be $0$ and $\pm p^k$ for some positive integer $k$ such that $\sqrt{p^{2^s} q} \leq p^k < q$.
\end{theorem}
One consequence \cite[Corollary 1.8]{Aubry-Katz-Langevin-15} of this is that $W_{F,d}$ can never be preferred three-valued if the degree of $F$ over its prime field is divisible by $4$.
\begin{corollary}
If $F$ is a finite field of characteristic $p$ with $4\mid [F:\F_p]$, and $d$ is an invertible exponent over $F$, then $W_{F,d}$ is not preferred three-valued.
\end{corollary}

We saw in Theorem \ref{Linda} that there are constraints on three-valued $W_{F,d}$ when the exponent $d$ becomes degenerate in some proper subfield of $F$.
This prompts the following definition.
\begin{definition}[Niho Exponent]
If $F$ is a finite field of characteristic $p$ with $[F:\F_p]$ even, then we say that an invertible exponent $d$ over $F$ is a {\it Niho exponent over $F$} to mean that $d$ is nondegenerate over $F$ but $d$ is degenerate in the unique subfield $K$ of $F$ with $[F:K]=2$.
\end{definition}
These exponents are named after Niho, who studied them in \cite{Niho}.
It turns out that Niho exponents never yield three-valued Weil sums.
\begin{theorem}\label{Barbara}
If $F$ is a finite field and $d$ is a Niho exponent over $F$, then $W_{F,d}$ is not three-valued.
\end{theorem}
The characteristic $p=2$ case was proved in \cite[Theorem 2]{Charpin}, and the general result was proved in \cite[Theorem 9]{Helleseth-Lahtonen-Rosendahl}.
For a different proof in odd characteristic, see \cite[\S 8]{Aubry-Katz-Langevin-15}.
One consequence of Theorems \ref{Sally}, \ref{Agnes}, and \ref{Barbara} is that Conjecture \ref{Thomas} must be true in prime fields (see \cite[Corollary 1.8]{Katz-12}) and their quadratic extensions.
\begin{corollary}
If $F$ is a finite field of characteristic $p$ and order $p$ or $p^2$, and if $d$ is an invertible exponent over $F$, then $W_{F,d}$ is not three-valued.
\end{corollary}
Otherwise Theorem \ref{Agnes} would force $d$ to be degenerate in the prime field $\F_p$, which would make $d$ either degenerate or Niho over $F$, according to which $W_{F,d}$ could not be three-valued by Theorems \ref{Sally} and \ref{Barbara}.

Many of the results on three-valued $W_{F,d}$ use results on $p$-divisibility (see Section \ref{Felix}).
The following result \cite[Theorem 1.5]{Katz-15} shows that being three-valued constrains the valuation of $W_{F,d}$.
\begin{theorem}\label{Victor}
Let $F$ be a field of characteristic $p$ and order $q=p^n$, and suppose that $d$ is an invertible exponent such that $W_{F,d}$ is three-valued with values $0$, $a$, and $b$.  Then one of the following holds:
\begin{enumerate}[(i)]
\item $v_p(a) > n/2$ and $v_p(b) > n/2$; or
\item\label{Justin} $v_p(a)=v_p(b)=n/2$, and $|a-b|$ is a power of $p$ with $|a-b| > \sqrt{q}$;
\end{enumerate}
and case \eqref{Justin} cannot occur if $p\in\{2,3\}$.
\end{theorem}
This result leads to a proof of Theorem \ref{Alexandra}, which verifies Conjecture \ref{Thomas} in characteristics $2$ and $3$: we cannot have a three-valued $W_{F,d}$ with $F$ of characteristic $p=2$ or $3$ and $[F:\F_p]$ a power of $2$, because if we could, then Theorem \ref{Julius}\eqref{Ferdinand} would make $V_{F,d} \leq [F:\F_p]/2$ while Theorem \ref{Victor} would make $V_{F,d} > [F:\F_p]/2$.
This result also constrains the possibilities for a counterexample to Conjecture \ref{Thomas} in characteristics $p \geq 5$ (see \cite[Remark 4.6]{Katz-15} and \cite[\S5]{Katz-Langevin-16}).
Of course the main open problem which encompasses all others in the realm of three-valued $W_{F,d}$ is whether Table \ref{Nancy} is in fact a complete listing of all three-valued $W_{F,d}$.
\begin{openproblem}
Is every pair $(F,d)$ of finite field and invertible exponent such that $W_{F,d}$ is three-valued accounted for (up to the equivalence of Definition \ref{Elizabeth}) on Table \ref{Nancy}?
\end{openproblem}
Clearly if this were answered affirmatively, then Conjectures \ref{Simon} and \ref{Thomas} would also be answered affirmatively.

We have mainly focused on three-valued $W_{F,d}$ since three is the minimum number of values possible when $d$ is nondegenerate over $F$.
There is also interest in $(F,d)$ pairs such that $W_{F,d}$ takes on few values, albeit not as few as three.
Various examples of four-valued Weil spectra arising from Niho exponents have been found, and a synthesis and generalization of these results for fields of characteristic $2$ can be found in \cite[Theorem 23]{Dobbertin-Felke-Helleseth-Rosendahl}.

\section{Presence of a Zero Value}\label{Nigel}

In this section, we concern ourselves with the question as to when the Weil spectrum $\{W_{F,d}(a): a \in F^*\}$ contains $0$.
If we had allowed $a$ to be $0$ in the definition of Weil spectrum, then this would be always be trivially true.
As a matter of fact, it is often the case that $W_{F,d}(a)=0$ for some nonzero $a$.
This prompted Helleseth to make the following conjecture (see \cite{Helleseth-71}, \cite[Conjecture 5.1]{Helleseth-76}).
\begin{conjecture}[Helleseth Vanishing Conjecture, 1971]\label{Rebecca}
If $F$ is a finite field of characteristic $p$ with $\card{F}>2$, and $d$ is an invertible exponent over $F$ with $d \equiv 1 \pmod{p-1}$, then $W_{F,d}(a)=0$ for some $a \in F^*$.
\end{conjecture}
The original conjecture did not contain the proviso that $\card{F}>2$, which is necessary to exclude a trivial exception.
This conjecture appears to be even more difficult to approach then Helleseth's Three-Valued conjecture (presented as Conjecture \ref{Thomas} in this paper).
Theorem \ref{Norbert} can be taken as a proof of Conjecture \ref{Rebecca} in the special case where $W_{F,d}$ is three-valued.
An equivalent formulation of Conjecture \ref{Rebecca} can be expressed in terms of the {\it Dedekind determinant},
\[
D_{F,d} = \prod_{a\in F^*} W_{F,d}(a),
\]
which leads to the following equivalent statement of Conjecture \ref{Rebecca}.
\begin{conjecture}[Helleseth 1971 Vanishing Conjecture, restated]
If $F$ is a finite field of characteristic $p$ with $\card{F}>2$, and $d$ is an invertible exponent over $F$ with $d \equiv 1 \pmod{p-1}$, then $D_{F,d}=0$.
\end{conjecture}
Although it has not been shown that $D_{F,d}$ always vanishes, some partial progress has been made by Aubry and Langevin \cite[Theorem 2]{Aubry-Langevin}, who show vanishing modulo certain primes.
\begin{theorem}
If $F$ is a finite field of characteristic $p$ and order $q=p^n > 2$, and $d$ is an invertible exponent over $F$ with $d \equiv 1 \pmod{p-1}$, then $D_{F,d} \equiv 0 \pmod{3}$.  Furthermore, if $n$ is a power of a prime $\ell$ with $\ell\nmid p-2$, then $D_{F,d} \equiv 0 \pmod{\ell}$.
\end{theorem}
Conjecture \ref{Rebecca} can also be expressed in terms of point counting.
If our finite field $F$ has order $q$, then Lemma \ref{Ursula} can be employed with $k=q-1$ and $\{b_1,\ldots,b_{q-1}\}=F^*$ to calculate
\[
D_{F,d}=\frac{q^2 N_{b_1,\ldots,b_{q-1}}-q^{q-1}}{(q-1)^2},
\]
where $N_{b_1,\ldots,b_{q-1}}$ is as defined to be the number of solutions $(x_1,\ldots,x_{q-1}) \in F^{q-1}$ to the system of equations
\begin{align}
\begin{split}\label{Andrew}
b_1 x_1 + \cdots + b_{q-1} x_{q-1} & = 0 \\
x_1^d+\cdots +x_{q-1}^d & =0.
\end{split}
\end{align}
This gives the following restated version of Conjecture \ref{Rebecca}, proposed in \cite[p.~38]{Helleseth-02}.
\begin{conjecture}[Helleseth's 1971 Vanishing Conjecture, restated again]
Let $F$ be a finite field of characteristic $p$ and order $q > 2$, and let $d$ be an invertible exponent over $F$ with $d \equiv 1 \pmod{p-1}$.
Let the elements of $F^*$ be written $b_1,\ldots,b_{q-1}$.
Then the number of solutions $(x_1,\ldots,x_{q-1}) \in F^{q-1}$ to the system of equations \eqref{Andrew} is equal to $q^{q-3}$.
\end{conjecture}
Thus the conjecture amounts to a precise determination of the cardinality of a hyperplane section of a Fermat variety.

\section*{Acknowledgement}

The author thanks Yakov Sapozhnikov, Arne Winterhof, and an anonymous reviewer, whose reading of and comments on the manuscript greatly helped the author.

redpajama_set_name: RedPajamaArXiv


Now that drupa 2016's over, we can start hoping that at drupa 2020 the environmental will be in sharper focus. This year's organisers missed a great opportunity to take the sustainability lead.
Drupa 2016's attendance and sales numbers show that print is in rude health. Investment into print technology is on the up, so why should drupa bother with promoting print's sustainability? The simple answer is to support exhibitors and the printing industry. And 2020 is an important year for European companies and citizens because it's when the European Union's current Environmental Action Plan (EAP) concludes. One of the plan's three key objectives is "to turn the Union into a resource-efficient, green, and competitive low-carbon economy". What a gift for print and drupa 2020!
The printing industry already supports this goal with extensive recycling of used printing plates and printed paper in most geographies around the world. Europeans recycle 74% of paper used. Japan labels print to tell consumers how it can be recycled. Printed matter has a one-time carbon footprint. Printing is one of the few industries with a dedicated ISO team developing environmental standards for printing companies and machinery manufacturers. Print has established supply chains for recycling and supports the notion of a circular economy on a massive scale. These are messages that can encourage wider use of print and more visitors to drupa, including agencies, designers and even consumers.
Messe Düsseldorf has a chance to take a global lead to demonstrate print's sustainability. For drupa 2020, how about random sustainability facts on hall signage: "Did you know that 98% of printing plates are recycled for use in building materials and vehicle manufacturing?" Or "Use and reuse paper because it can be recycled more than five times!" Improving waste handling advice would be good too. At this year's show there was virtually no additional emphasis for visitors on how they can recycle all those print samples. Bins at the exits perhaps? You get the idea.
A more radical suggestion is to give exhibitors the opportunity to print what the local community wants, instead of random samples that have to go into the waste stream. Messe Düsseldorf could start an outreach programme to work with publishers to produce textbooks and the like that will have value for reuse. We spoke to a number of exhibitors at drupa 2016 who liked the idea of this.
The Messe could also consider promoting its own environmental credentials, based on drupa 2020's energy use, and measures for managing emissions. A series of awareness seminars for brand owners and consumers might also support print. Print is often taken for granted as an old medium, with a heavy environmental tread. Why not position it as the trendy and cool alternative to digital media, the medium that uniquely supports the three Rs: reduce, reuse, recycle.
In case you missed it, Rich mineral stone paper had a stand at Drupa supporting sustainability and environment in Print.
While not directly answering your points, and something that they made no fuss of themselves, Heidelberg for one has offset the carbon footprint of its stand. It will plant 10,700 trees in Togo at a project it has supported over some years. All the equipment on the stand was labelled co2 neutral. Others may have done something similar.


text:
Purwakarta kan syfta på följande platser:

Indonesien

 Kabupaten Purwakarta, kabupaten, Jawa Barat,
 Purwakarta (ort i Indonesien), Jawa Barat,
 Kecamatan Purwakarta, distrikt, Jawa Barat,

Robotskapade Indonesienförgreningar

redpajama_set_name: RedPajamaWikipedia


text:
class CSVAdapter(object):
    def __init__(self, db, **kwargs):
        self._connection_opts = kwargs
        self._connection = None
        self._db = db

    def select_all(self, mapping, obj_class):
        reader = self._db.reader(self.connection)
        for row in reader:
            obj = obj_class()
            yield mapping.row_to_object(row, obj)

    @property
    def connection(self):
        return self._connection if self._connection else self.connect()

    def connect(self):
        if not self._connection:
            self._connection = self._db.connect(**self._connection_opts)
        return self._connection

    def close_connection(self):
        if self._connection:
            self._connection.close()
            self._connection = None

redpajama_set_name: RedPajamaGithub


What if Augmented Reality is Audio?
↑ Meet Detour, a location-aware audio guide.
What if we've been all wrong about Augmented Reality? What if it isn't some intrusive pair of glasses that mediates your entire field of vision? What if it's simply audio?
It makes sense. What is the radio in your car if not a way to augment your reality while driving? A kid on the streetcar puts on his headphones and disappears. We take the power of audio to augment our realities for granted.
Now we can take what we've been doing with audio and pair it with location awareness. Technological evolution often happens when you pair things and get something greater than the sum of those parts.
For example: One of Detour's audio tours is narrated by a fictional character (Ulrich Fürst, a severe German philosopher with a sweet tooth).
Imagine what could be done with location-aware audio and fiction. It could be a whole new way for the reader to get into the mind of the writer—walking around in the world of the story.


text:
Emilio Samayoa (born 22 July 1964) is a Guatemalan sprinter. He competed in the men's 100 metres at the 1984 Summer Olympics.

References

1964 births
Living people
Athletes (track and field) at the 1984 Summer Olympics
Guatemalan male sprinters
Olympic athletes of Guatemala
Place of birth missing (living people)


text:
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Looking to enjoy Perth's coastal lifestyle? Well, we're planning an exciting new community called Alkimos Vista. And we want the shapers, makers and co-creators that will make it thrive.
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text:
Host a Presidential Game Party
June 26 - July 4, 2015 - Presidential Game Party
Do you think you have what it takes to be the next President? Now is the time to get your game on with a product perfect for the upcoming 4th of July holiday!
The Presidential Game has an outstanding 4.9 out of 5 star user rating on Amazon and is great for adults and students alike. Battle in teams as Republications vs. Democrats (or Red vs. Blue!) to see who can win the most electoral votes to earn the nomination. Not only will you enjoy the strategy involved but those who participate will walk away with a much better understanding of how our electoral system works.
Tryazon will choose 50 hosts for this party opportunity. Those picked will have the opportunity to receive a party pack with The Presidential Game to showcase in June or July! Learn more about the game and party by clicking the resources below.
Party Pack Products
Hosts selected for this party will receive a party pack containing the following items:
• The Presidential Game
• Exclusive coupons for you and your guests!
• Party Host Guide - to help you plan and carry out the party
The Presidential Game has won a number of awards for it's excellence as an educational tool in teaching about the electoral process. It has been featured in The Wall Street Journal and is still a The National Parenting Center featured favorite.
Known as the most powerful game in the world, it is based on strategy with the Democrats against the Republicans in an epic battle for the Electoral votes in each state. Whether you are an adult or a student, this game is sure to provide hours of educational fun. Even if you don't know exactly how the Electoral College works, this game is still easy and exciting to play. The Presidential game is an innovative and engaging way to teach kids about the political process. Whether you play at home or buy for classroom use, this game is a must-have. To start game play, the two teams (Democrats and Republicans) decide whether they want to campaign or fundraise. A quick roll of the dice and political cards are assigned to each team based on the scenario they selected. Ultimately, whichever team gets 270 or more Electoral Votes wins the game. For easier record keeping, an interactive Electoral calculator is included so votes can be tracked on any computer, tablet or smartphone. Simply enter the access code found on the back of the game board. This game is recommended for ages 11 years and older.
Use #tryazon and #thepresidentialgame to include your tweet on this page!
Tweets about #tryazon, #thepresidentialgame
Note: These timelines may not show up on certain browsers
The application period has now ended. Please check out our other Featured Party Opportunities.
May 29Applications Due
All party host applications need to be received by this date.
June 2Hosts Selected
Applicants will receive notification by email by this date. Those party hosts who are accepted and confirm, will have their party packs shipped to them.
June 3Start Party Planning
Hosts can begin planning for their parties. Tryazon will provide party guides and resources to assist!
June 26Host Parties
The fun begins in late June, when parties will be hosted.
Number of Host Spots
Tryazon will select up to 50 hosts for this party.
The Presidential is a fun, easy to learn game that is a cool way to introduce kids to politics and the electoral college...One capture[s] as many states to become president and the online interactive feature is great! Some strategy is involved as well making it an entertaining and educational family game.
Amazon Reviewer


Robert Gallery. The veteran left guard who was signed in free agency didn't just return to the starting lineup in Sunday's game against the Browns in Cleveland, he looked like a different player.
More active. More agile. More physical.
That's when Gallery had surgery to repair a groin injury, a procedure that forced him to miss three games and the team's bye-week practices. If his performance against the Browns was any indication, the wait to get him back was worth it.
"He had an existing issue that … was hampering him and he was playing through it," Carroll said, before repeating, "He looked better.
Ware, Allen, Babin, Wake, Woodley and Matthews were voted to the Pro Bowl either last season or in 2009 – or both.
Speaking of the Pro Bowl and Clemons, fan voting for this season has begun at NFL.com. You can cast your vote here.
The players return from their "off" day on Wednesday to begin preparing for Sunday's game against the Cincinnati Bengals at CenturyLink Field. Wednesday's practice will be pivotal for several players who are trying to return from injuries: quarterback Tarvaris Jackson (pectoral), center Max Unger (foot), running back Marshawn Lynch (back) and cornerback Roy Lewis (knee).
We also have a look at Sunday's practice in words, pictures and video. And, there's also the recap of how Whitehurst defended his title in the homerun derby after practice.
Here at Seahawks.com, we continues our series of articles on the players voted to the 35th Anniversary team with a look at one of the best – left tackle Walter Jones. But who is the best player in franchise history? We've also got a poll, where you can vote for Jones, or Steve Largent, or Cortez Kennedy, or Shaun Alexander, or Kenny Easley, or Jacob Green.
Clint Gresham. 'Tis the season. No, that was last week. But for the Seahawks' long snapper, it's always the giving season.
On Christmas Eve, Gresham donated five Wii systems to the families at Vision House, which provides transitional housing for homeless single mothers and their children. He returned on Wednesday, with autographed mini Seahawks helmets for the kids and then them outside for a few pointers on the fine art of snapping.
"I've been wanting to get involved somehow in the community, and just have an opportunity to share my faith and be involved in the lives of these kids," Gresham said today, after the team's next-to-last practice for Sunday's big game against the St. Louis Rams at Qwest Field to determine the champion of the NFC West.
The plan was to get first-round draft choice Russell Okung back this week, after he missed the previous two games with a sprained left ankle. But the rookie was very limited in practice the past two days and is listed as questionable, so Polumbus could make his fifth start.
"Ever since I called him Troy, he's been pretty good," Carroll cracked today after practice – the final full workout before Sunday's game in Arizona.
That was way back in September, when Carroll slipped and referred to Tyler as Troy – and Polumbus started the first three games at left tackle because Okung was out with a sprained right ankle.
Quinton Ganther, through the smiling eyes of Marshawn Lynch. Ganther – the veteran, and versatile, running back – hasn't just returned, he's among old friends.
Ganther, who was re-signed Tuesday, had been released Sept. 28 – eight days before Lynch was acquired in the trade with the Buffalo Bills. They grew up together in Oakland and played on rival Pop Warner teams.
"I was 11 or 12, and Quinton was like 30," Lynch cracked in the locker room before today's practice – the first to prepare for Sunday's game against the Raiders in Oakland.
ST. LOUIS – Greetings from the Edwards Jones Dome, where the Seahawks are starting to take the field to begin preparing for today's game against the Rams on the first Sunday of October.
The Seahawks are 2-1 to share the NFC West lead with the defending division champion Arizona Cardinals. They could be 3-1 and alone in first place entering the their bye week after today's action, because the Cardinals play the Chargers in San Diego and the Seahawks are facing a team they've beaten 10 consecutive times – including five in a row in the dome.
Could, however, won't translate to will unless the Seahawks take care of business.
The Rams have yielded 17, 16 and 16 points in their first three games, despite ranking 24th in total defense – 25th against the run and 24th against the pass. But they have forced eight turnovers (three interceptions and five fumble recoveries), and it was turnovers that greased the way to the Seahawks' loss to the Broncos in Denver two weeks ago.
Hasselbeck has started eight of the Seahawks' 10 consecutive wins and is 10-3 in his career as a starter against the Rams – including three-touchdown pass outings in 2006 and 2009 and a pair of 300-yard passing performances.
The Seahawks, of course, can help Hasselbeck by continuing to improve their running game. Today, Justin Forsett likely will be running behind yet another new line combination – one that is expected to feature the regular-season debut of first-round draft choice Russell Okung at left tackle and Tyler Polumbus stepping in for a sore-kneed Sean Locklear at right tackle. Polumbus started the first three games on the left side while Okung was out with a high ankle sprain.
It's not likely that the Seahawks' defense will have to face Steven Jackson today. He left last week's win over the Redskins with a strained groin and did not practice this week. Even if he does play, Jackson will not at full battering-ram strength.
That means the Seahawks' fifth-ranked run defense will see a lot of Kenneth Darby. He scored not only his first NFL rushing touchdown last week but his first since his 2005, when he was a junior at Alabama. It's not that Darby didn't play much for the Crimson Tide; it's that he was replaced when they got near the goal line. In fact, Darby finished his career at Alabama 241 yards shy of the school rushing record that was set by – you guessed it – Shaun Alexander.
Whether it's Darby, Jackson or a combination of the two, the Seahawks need to continue stuffing the run to put rookie QB Sam Bradford into passing situations. But even that is only part of their equation for success, because they then have to pressure the first pick overall in the April draft into making mistakes.
The Seahawks have seven sacks and 21 QB hits. All but one in each category, however, came in their two games at Qwest Field. They didn't get enough pressure on the Broncos' Kyle Orton and he completed 25 of 35 passes for 307 yards and two TDs.
We'll be back shortly with the list of inactive players.
Craig Terrill. The Seahawks kicked off October by re-signing the sack-producing/kick-blocking/guitar-playing defensive tackle.
Terrill was released Sept. 7, when versatile defensive back Jordan Babineaux was re-signed. To clear a roster spot for Terrill's return, linebacker Leroy Hill was placed on injured reserve because he will need surgery to repair the Achilles/calf injury he got in the Week 2 game against the Broncos in Denver.
"It's awesome," said Terrill, who got a phone call from the club this morning and was signed in time to participate in the final full practice before Sunday's game in St. Louis against the Rams.
Terrill was a fifth-round draft choice in 2004 and played in 76 games the past six seasons. He has blocked five field goals in his career, which ranks second in club history behind Joe Nash's eight; and had a three-sack game against the Oakland Raiders in 2006. He also has released a CD, CT, and sang the national anthem before the 2007 playoff game against the Dallas Cowboys at Qwest Field.
During his down time, Terrill was helping his wife, Rachel, raise their 2-year-old daughter, Journey.
That didn't take long, as Terrill worked on the scout team defensive line against the Seahawks offense in practice and also got a few snaps with the No. 1 D-line.


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I have very little hipster cred, so I'm going take the opportunity to do the hipster thing here and say that I knew about the Bell Witch before it was cool. In 1989, my folks and I were driving from Chicago to visit our family in Florida (remember them?). As we passed into Tennessee, we passed a sign that said, "See the Bell Witch Cave" or words to that effect. My folks were always up for a cave (and for a good supernatural story). So, long before An American Haunting, or Bell Witch: The Movie, we heard the story from the current owners of the cave.
I was looking through the photo album of this trip and saw pictures of a cave. "I wonder if that was the Bell Witch Cave. It would be about the right timing for that." So I searched for 'Bell Witch Cave' on Google Images and saw a (considerably less overexposed) shot of this same location on someone else's website. So I can say with about 90% certainty that this is, in fact, a picture of the Bell Witch Cave.
According to the legend, in the early 19th century, the family of a farmer named John Bell began to experience something that was generally thought of as supernatural. He, his family, and visitors to his home, heard voices. Sometimes it was the voice of a woman, at other times it was the voices of other people. It was reported that the voices, at one point, began repeating the words of two church services taking place simultaneously in two different churches miles from the Bell home.
The entity claimed to be "Kate Batts's witch." Kate Batts was a neighbor that the Bells had had problems with over some kind of economic transaction, either the purchase of land or of slaves. Given the time period that this story took place in, my money's on the latter. At any rate, the apparition was given the name "Kate" and would apparently respond to that name.
In the end, "Kate," presumably the spirit and not the neighbor (though in my memory, it seemed that the man telling the story was unclear on this point), fled to the cave. There are several legends of her interacting with people in the cave.
Do I believe in the Bell Witch? I try to keep an open mind about things like ghosts, because I have seen some things that seem unexplainable (and on several occasions I was by myself, so they couldn't have been practical jokes or anything of that nature). But I do wonder if the Bell Witch was real or was an attempt to slander a neighbor who had a grievance.
Next up: Stone Mountain, Atlanta. I may have to see if there are any public domain photos of the park because I don't have a single one in my collection.
And the pictures he took really are stunning. I think that my personal favorite is the tail end of an RV with mountainous scenery as it drives past a view of the Badlands in South Dakota.
This article talks about the latest developments in forensic science. One of the techniques covered is what's called genetic phenotyping, where they can now take a DNA sample and pull out hair, eye, and skin color and, often traits like whether the source of the sample had freckles. They can even sometimes get a general idea of the shape of the person's face.
Of course, the hair thing might not be foolproof, as (totally aside from the existence of hair dyes), people do lose their hair sometimes and hair does eventually gray. In fact, I knew two young men in my youth who lost their hair at very young ages. It's likely that there might be some kind of genetic component to the hair loss, but statistically speaking, in their teens or 20s the reconstructions would probably have shown them with full heads of hair.
The photograph of the photographer, by the way, was created by DNA phenotyping. There's an interactive feature on the website where you can compare that image to actual photographs of actual photographers to see if you can figure out which was is Max.
Well, it's been a while since we've had some unrest in Africa, so I guess we're due. And since we're talking about parks in the magazine this year, this article is a "twofer," part of the Power of Parks series and about unrest in Africa.
Virunga National Park in the Democratic Republic of the Congo is the oldest national park in Africa. It is also part of some kind of turf war between at least two militias. The Battle for Virunga covers some of this recent history and discusses some of the things that the (at the moment) current director, Emmanuel de Merode, is doing to improve the park. One of these things is that they are building hydroelectric power plants in the park, hoping that the electricity being produced will (a) cover the park's expenses into the future and (b) give potential entrepreneurs the chance to begin to develop businesses in the region that will give the children of the area something to aspire to besides joining a militia.
As we go farther and farther back, the "ripped from the headlines" nature of some of these articles is blunted a bit. Can't wait until we take a "look forward" at what will happen to Cuba under Castro and things of that nature. Today we look back at the developments in Libya after the 2011 death of Muammar Gaddafi.
Prior to Libya's independence in 1950, Libya had previously been run by the United Nations, then prior to that, by Italy, and then prior to *that* by the Ottoman Empire and then prior to that by Rome. In fact, emperor Septimius Severus had been born in Libya, in a city known as Leptus Magna. When Gaddafi took power in 2011, he disdained all of this history, particularly the parts where the country had been ruled by Rome and Italy.
In this article, we see a picture of Libya in very late 2012 as a country that is moving both toward its future while trying to recapture the past that Gaddafi tried to suppress. We see the unrest that still existed in late 2012, but we also see people going on with their lives, hopeful that they will have a future.
And, of course, as we know now, the first war in 2011 that led to the fall and death of Gaddafi, was followed by a second war that continues, well, at least until I'm writing this in 2016. Now Libya his hemorrhaging people, with thousands of people fleeing every year.
I'm hoping to start studying Vietnamese in 2017, because it's one of the languages that I have to use the translation service for most in my job. It looks like I may also need to learn Arabic because immigrants from Arabic-speaking nations are on the rise here, as well. My side of town is where groups like Catholic Charities like to resettle the refugees because services are easy to access in this area.
In The Bite that Heals, Holland takes us to speak with scientists who are making medication from venom. We start in Mexico, where a man named Michael was healed of his ankylosing spondylitis apparently by a scorpion sting. At the time this article went to press, about a dozen medications had been developed from venom, including the blood pressure medication captopril. The web page of Zoltan Takacs, one of the scientists that Holland speaks with, has a list of (at this time) around 15 medications that are currently being sold that are derived from venom. This list has an additional five that are in clinical trials.


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Q: View, edit and update data (from C++ ) in QML with multiple views, while the Data stays in C++ (subscribe to data) I have some data stored in instances of a C++ class (Data.cpp). Now i want to be able to view and edit this data from 2 seperate representations in QML, so that if the values in View1 are changed, the data itself (C++) is changed and the value displayed by View2 as well (because it gets notified when the C++ data changes).
Here is what I got so far:
Data.h
class Data : public QObject
{
    Q_OBJECT
    Q_PROPERTY(QString name READ name WRITE setName NOTIFY nameChanged)

public:
    Data(std::string name);
    QString name();
    void setName(const QString &n);

signals:
    void nameChanged();

private:
    std::string _name;
};

Parser.h (provides a list of Data)
class Parser : public QObject
{
    Q_OBJECT
    Q_PROPERTY(QList<QObject*> list READ list NOTIFY listChanged)
    //QList<Data*> is not working with QML :(

public:
    Parser(QObject *parent = 0);
    QList<QObject*> list() //stuff below is implementd in Parser.cpp
    {
       _list.append(new Data("name 1"));
       _list.append(new Data("name 2"));
       _list.append(new Data("name 3"));
        return _list;
    }

signals:
    void listChanged();

private:
    QList<QObject*> _list;
};

QML part:
    ListView
    {
        id: view1
        anchors.fill: parent
        spacing: 5
        delegate: Text { text: name}
        model: ListModel{Component.onCompleted: getModel()}
    }
    ListView
    {
        id: list2
        anchors.fill: parent
        spacing: 5
        delegate: Text { text: name}
        model: ListModel{Component.onCompleted: getModel()}
    }
    function getModel()
    {
        var m = parser.list;
        for(var i=0; i<m.length; i++)
        {
            list.model.append(m[i]); //simply returning the list (m) does not work
        }
    }

Now if I click on an item in view1 (for example) i want the name of the corresponding Data to change, and the name displayed in view2 accordingly. If I modified the name from C++, the new name should be displayed in both views.
Is there any way to do this? I'm stuck on this for days...
Thanks for your help.

EDIT:
I asked a more specific question to this topic here.

A: It's very much possible, but you've got a few problems:

*

*You want to expose your list of Data objects as a QQmlListProperty. This is the proper way to get lists into QML

*If your list is properly exposed as a QQmlListProperty you can just set it as the model of your ListView without doing that weird getModel() hack that you're doing now

*You shouldn't be added items to your list in the getter or else you'll be appending to it every time QML tries to read it.

Once that's resolved, you can update a Data object simply by interacting with the reference to the current model item in your delegate.
Here's a full working example. I've added a MouseArea in the QML that changes the model and also a timer in C++ that also changes the model, to show that changes to either side are instantly reflected in the UI.
main.cpp:
#include <QGuiApplication>
#include <QtQuick>

class Data : public QObject
{
    Q_OBJECT
    Q_PROPERTY(QString name READ name WRITE setName NOTIFY nameChanged)

public:
    Data(const QString &n) : _name(n) { }
    QString name() const { return _name; }

    void setName(const QString &n) {
        if (_name == n)
            return;
        _name = n;
        emit nameChanged(n);
    }

signals:
    void nameChanged(const QString &n);

private:
    QString _name;
};

class Parser : public QObject
{
    Q_OBJECT
    Q_PROPERTY(QQmlListProperty<Data> list READ list CONSTANT)

public:
    Parser(QObject *parent = 0) {
        _list.append(new Data(QStringLiteral("name 1")));
        _list.append(new Data(QStringLiteral("name 2")));
        _list.append(new Data(QStringLiteral("name 3")));

        startTimer(5000);
    }

    QQmlListProperty<Data> list() {
        return QQmlListProperty<Data>(this, _list);
    }

    void timerEvent(QTimerEvent *) {
        _list[1]->setName(_list[1]->name() + QStringLiteral("C++"));
    }

private:
    QList<Data*> _list;
};

int main(int argc, char *argv[])
{
    QGuiApplication a(argc, argv);

    qmlRegisterType<Data>();

    QQuickView view;
    view.rootContext()->setContextProperty(QStringLiteral("parser"), new Parser);
    view.setSource(QUrl("qrc:///qml/main.qml"));
    view.showNormal();

    return a.exec();
}

#include "main.moc"

main.qml:
import QtQuick 2.0

Rectangle {
    width: 800
    height: 600

    ListView {
        id: view1
        anchors { top: parent.top; left: parent.left; bottom: parent.bottom }
        width: parent.width / 2
        spacing: 5

        delegate: Item {
            height: 30
            width: parent.width

            Text { text: name }

            MouseArea {
                anchors.fill: parent
                onClicked: model.name += "1";
            }
        }
        model: parser.list
    }

    ListView {
        id: view2
        anchors { top: parent.top; right: parent.right; bottom: parent.bottom }
        width: parent.width / 2
        spacing: 5

        delegate: Item {
            height: 30
            width: parent.width

            Text { text: name }

            MouseArea {
                anchors.fill: parent
                onClicked: model.name += "2";
            }
        }
        model: parser.list
    }
}

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