[en] Let $P$ and $Q$ be two non-zero multiplicatively independent polynomials with coefficients in a finite field $\mathbb{F}$. Adapting a result of R.~Villemaire, we show that multiplication of polynomials is a ternary relation $\{(A,B,C)\in\mathbb{F}[X]\mid A.B=C\}$ definable by a first-order formula in a suitable structure containing both functions $V_P$ and $V_Q$ where $V_A(B)$ is defined as the greatest power of $A$ dividing $B$. Such a result has to be considered in the context of a possible analogue of Cobham's theorem for sets of polynomials whose $P$-expansions are recognized by some finite automaton.
Disciplines :
Mathematics
Author, co-author :
Rigo, Michel ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Waxweiler, Laurent
Language :
English
Title :
Defining multiplication for polynomials over a finite field.