The Cayley-Menger determinant is irreducible for n≥3n\geq 3

We prove that the Cayley-Menger determinant of an $n$-dimensional simplex is an absolutely irreducible polynomial for $n\geq3.$ We also study the irreducibility of polynomials associated to related geometric constructions.Comment: 7 pages, 4 figure

Factoring bivariate sparse (lacunary) polynomials

We present a deterministic algorithm for computing all irreducible factors of degree $\le d$ of a given bivariate polynomial $f\in K[x,y]$ over an algebraic number field $K$ and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in $d$. Moreover, we show that the factors over \Qbarra of degree $\le d$ which are not binomials can also be computed in time polynomial in the sparse length of the input and in $d$.Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a multivariate version of Theorem 1 had independently been achieved by Erich Kaltofen and Pascal Koira

Intrinsic palindromic numbers

We introduce a notion of palindromicity of a natural number which is independent of the base. We study the existence and density of palindromic and multiple palindromic numbers, and we raise several related questions.Comment: 6 pages, Latex2

Successive minima of projective toric varieties

We compute the successive minima of the projective toric variety X_\cA associated to a finite set \cA \subset \Z^n. As a consequence of this computation and of the results of S.-W. Zhang on the distribution of small points, we derive estimates for the height of the subvariety X_\cA and of the \cA-resultant. These estimates allow us to obtain an arithmetic analogue of the Bezout-Kushnirenko's theorem concerning the number of solutions of a system of polynomial equations. As an application of this result, we improve the known estimates for the height of the polynomials in the sparse Nullstellensatz.Comment: Revised version. In French, 25 p