References of "Waxweiler, Laurent"
     in
Bookmark and Share    
Full Text
See detailDefining multiplication for polynomials over a finite field.
Rigo, Michel ULg; Waxweiler, Laurent

E-print/Working paper (2011)

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 ... [more ▼]

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. [less ▲]

Detailed reference viewed: 52 (4 ULg)
Full Text
Peer Reviewed
See detailLogical characterization of recognizable sets of polynomials over a finite field
Rigo, Michel ULg; Waxweiler, Laurent

in International Journal of Foundations of Computer Science (2011), 22(7), 1549-1563

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination ... [more ▼]

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study $P$-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory. [less ▲]

Detailed reference viewed: 78 (13 ULg)