arXiv subjects
Publications and source records attributed to Yves Aubry.
We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $α$ such that $K(α)$ is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.
A graph $G$ is free $(a,b)$-choosable if for any vertex $v$ with $b$ colors assigned and for any list of colors of size $a$ associated with each vertex $u\ne v$, the coloring can be completed by choosing for $u$ a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. In this note, a necessary and sufficient condition for a cycle to be free $(a,b)$-choosable is given. As a corollary, some choosability results are derived for graphs in which cycles are connected by a tree structure.
We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.
We are concern about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs cross-correlations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction.
We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.
For a graph $G$ with a given list assignment $L$ on the vertices, we give an algebraical description of the set of all weights $w$ such that $G$ is $(L,w)$-colorable, called permissible weights. Moreover, for a graph $G$ with a given list $L$ and a given permissible weight $w$, we describe the set of all $(L,w)$-colorings of $G$. By the way, we solve the {\sl channel assignment problem}. Furthermore, we describe the set of solutions to the {\sl on call problem}: when $w$ is not a permissible weight, we find all the nearest permissible weights $w'$. Finally, we give a solution to the non-recoloring problem keeping a given subcoloring.
A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertex, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer $m\ge 1$, every finite triangle-free induced subgraph of the triangular lattice is $(5m,2m)$-choosable.
A graph $G$ with a list of colors $L(v)$ and weight $w(v)$ for each vertex $v$ is $(L,w)$-colorable if one can choose a subset of $w(v)$ colors from $L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be $(L,w)$-colorable for some list assignments $L$. Furthermore, we solve the problem of the free-choosability of a cycle.
A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertices, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the $(a,b)$-choosability of a graph and the $(a,b)$-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove the $(5,2)$-choosability and $(7,3)$-colorability of triangle-free induced subgraphs of the triangular lattice.
We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition are functions that are not APN on infinitely many extensions of ${\bf F}_{2^m}$. Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold member ($2^k+1$) or a Kasami-Welch number ($4^k-2^k+1$). We also have partial results on functions of even degree, and functions that have degree $2^k+1$.
We give a geometric characterization of vectorial boolean functions with differential uniformity less or equal to 4.