Distribution of values of polynomial Fermat quotients
In analogy with the Fermat quotient \((a^{p-1}-1)/p\pmod p\) defined for primes \(p\) and integers \(a\) coprime to \(p\), the authors define for an irreducible polynomial \(P\in {\mathbb F}_q[X]\) of degree \(n\geq 2\) and a polynomial \(A\in {\mathbb F}_q[X]\) of degree at most \(n-1\) the quotient \(q_P(A)\equiv (A^{q^n-1}-1)/P\pmod P\). Here \(q=p^r\) is a power of the prime \(p\). In the paper under review, the authors study the number of fixed points and the image size of the map \(q_P(A)\) when \(P\) is fixed and \(A\) varies in the set \({\mathcal P}_{n,q}\) of polynomials of degree at most \(n-1\) in \({\mathbb F}_q[X]\). Denoting by \(N_P(n,q)\) for the number of such fixed points, the authors prove that \[ N_P(n,q)\leq \left\{ \begin{matrix} 1+c_q\frac{q^{n-\lceil n/p\rceil-1}-1}{q-1} & n=2,~3,~{\text{or}}~n\geq 4~{\text{and}}~q\in \{2,4\},\\ q^{n/2}+1/2, & n\geq 4~{\text{and}}~q~{\text{is~odd}},\\ 2^n, & n\geq 4~{\text{and}}~q\geq 8~{\text{is~even}}. \end{matrix}\right. \] Here, \(c_q=2\) if \(q\) is odd and \(c_q=1\) if \(q\) is even. Denoting \(M_P(n,q)\) for the image size of \(q_P(A)\) when \(A\) varies in \({\mathcal P}_{n,q}\), they prove that \[ 1+\frac{q^n-q^{\lceil n/p\rceil}}{q-1}\geq {\mathcal M}_P(n,q)\geq \left\{ \begin{matrix} q^{\lceil 3/p\rceil}+1, & n=3,\\ (1+o(1)) \frac{q^{n-2}(q-1)}{n(n-1)}, & n\geq 4\end{matrix}\right. \] as \(q^n\to\infty\). The classical ``Prime Number Theorem'' in this context, which is the asymptotic estimate \(q^k/k\) for the number of irreducible monic polynomials of degree \(k\) in \({\mathbb F}_q[X]\) plays an important role in some of the proofs.
scientific article; zbMATH DE number 6125342