  Let $\Bbb F_q$ be a finite field with $\text{char}\,\Bbb F_q=p$ and $n>0$ an
integer with $\text{gcd}(n, \log_pq)=1$. Let $(\ )^*:\Bbb F_q({\tt
x}_0,\dots,{\tt x}_{n-1})\to\Bbb F_q({\tt x}_0,\dots,{\tt x}_{n-1})$ be the
$\Bbb F_q$-monomorphism defined by ${\tt x}_i^*={\tt x}_{i+1}$ for $0\le i<
n-1$ and ${\tt x}_{n-1}^*={\tt x}_0^q$. For $f,g\in\Bbb F_q({\tt
x}_0,\dots,{\tt x}_{n-1})\setminus\Bbb F_q$, define $f\circ
g=f(g,g^*,\dots,g^{(n-1)*})$. Then $(\Bbb F_q({\tt x}_0,\dots,{\tt
x}_{n-1})\setminus\Bbb F_q,\,\circ)$ is a monoid whose invertible elements are
called global $\mathcal P$-forms. Global $\mathcal P$-forms were first
introduced by H. Dobbertin in 2001 with $q=2$ to study certain type of
permutation polynomials of $\Bbb F_{2^m}$ with $\text{gcd}(m,n)=1$; global
$\mathcal P$-forms with $q=p$ for an arbitrary prime $p$ were considered by W.
More in 2005. In this paper, we discuss some fundamental questions about global
$\mathcal P$-forms, some of which are answered and others remain open.

Number Theory;Group Theory