We provide a formula for the Ehrhart polynomial of connected matroid size $n$ and rank $k$ with least number bases, also known as minimal matroid. prove that their polytopes are positive $h^*$-real-rooted (and hence unimodal). operation circuit-hyperplane relaxation relates matroids subdivisions, preserves positivity. state two conjectures: indeed all $h^*$-real-rooted, coefficients fixed cardi...

We want to achieve efficient exact computations, such as the rank, of sparse matrices over finite fields. We therefore compare the practical behaviors, on a wide range of sparse matrices of the deterministic Gaussian elimination technique, using reordering heuristics, with the probabilistic, blackbox, Wiedemann algorithm. Indeed, we prove here that the latter is the fastest iterative variant of...

We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain D. This theorem is relative to any element of D. The ingredients are: the arithmetic of Laurent polynomials over D, a recursive ’index function’ and simple mathematical induction. Taking reciprocals gives a ’Berlekamp-Massey theorem’ i.e. a recursive construction of the polynomials arising ...

The diagonal of a multivariate power series F is the univariate power series Diag F generated by the diagonal terms of F . Diagonals form an important class of power series; they occur frequently in number theory, theoretical physics and enumerative combinatorics. We study algorithmic questions related to diagonals in the case where F is the Taylor expansion of a bivariate rational function. It...

In this work we analyse the GLVmethod of Gallant, Lambert and Vanstone (CRYPTO 2001) which uses a fast endomorphism Φ with minimal polynomial X + rX+ s to compute any multiple kP of a point P of order n lying on an elliptic curve. First we fill in a gap in the proof of the bound of the kernel K vectors of the reduction map f : (i, j) 7→ i+λj (mod n). In particular, we prove the GLV decompositio...