Let $f_i$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_i$ such that $\sum g_i{f}_i=1$. The effective versions of this result bound the degrees of the $g_i$ in terms of the degrees of the $f_j$. The aim of this paper is to generalize this to the case when the $f_i$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

On montre comment écrire de grandes familles, avec de hautes multiplicités, de cas d’égalité $A+B=C$ pour l’inégalité de Stothers-Mason (si $A\left(X\right),B\left(X\right),C\left(X\right)$ sont des polynômes premiers entre eux, le nombre exact de racines du produit $ABC$ dépasse de $1$ le plus grand des degrés des composantes $A,B,C)$. On développera pour cela des techniques polynomiales itératives inspirées des décompositions de Dunford-Schwartz et de fonctions de Belyi. Des exemples d’application avec les conjectures $\left(abc\right)$ ou de M. Hall sont développés.

1. Introduction. The study of divisor functions of matrices arose legitimately in the context of arithmetic of matrices, and the question of the number of (possibly weighted) inequivalent factorizations of an integer matrix was asked. However, till now only partial answers were available. Nanda [6] evaluated the case of prime matrices and Narang [7] gave an evaluation for 2×2 matrices. We obtained a recursion in the size of the matrices and the weights of the divisors [1,2] which helped us obtain...

We consider inequalities between sums of monomials that hold for all p-Newton sequences. This continues recent work in which inequalities between sums of two, two-term monomials were combinatorially characterized (via the indices involved). Our focus is on the case of sums of three, two-term monomials, but this is very much more complicated. We develop and use a theory of exponential polynomial inequalities to give a sufficient condition for general monomial sum inequalities, and use the sufficient...

We prove that there are absolute constants $c_1\>0$ and $c_2\>0$ such that for every$$\{a_0,a_1,...,a_n\}\subset [1,M],1\le M\le exp\left(c_1{n}^{1/4}\right),$$there are$$b_0,b_1,...,b_n\in \{-1,0,1\}$$such that$$P\left(z\right)=\sum _{j=0}^n{b}_j{a}_j{z}^j$$has at least $c_2{n}^{1/4}$ distinct sign changes in $(0,1)$. This improves and extends earlier results of Bloch and Pólya.