  The breakthrough paper of Croot, Lev, Pach \cite{CLP} on progression-free
sets in $\Z_4^n$ introduced a polynomial method that has generated a wealth of
applications, such as Ellenberg and Gijswijt's solutions to the cap set problem
\cite{EG}. Using this method, we bound the size of a set of polynomials over
$\F_q$ of degree less than $n$ that is free of solutions to the equation
$\sum_{i=1}^k a_if_i^r=0$, where the coefficients $a_i$ are polynomials that
sum to 0 and the number of variables satisfies $k\geq 2r^2+1$. The bound we
obtain is of the form $q^{cn}$ for some constant $c<1$. This is in contrast to
the best bounds known for the corresponding problem in the integers, which
offer only a logarithmic saving, but work already with as few as $k\geq r^2+1$
variables.

Combinatorics;Number Theory