Reference : Defining multiplication for polynomials over a finite field.
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Physical, chemical, mathematical & earth Sciences : Mathematics
Defining multiplication for polynomials over a finite field.
Rigo, Michel mailto [Université de Liège - ULg > Département de mathématique > Mathématiques discrètes >]
Waxweiler, Laurent [> >]
[en] First order logic ; Finite Field ; definability
[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.

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