相关论文: Lemme d'Artin-Rees, theoreme d'Izumi et fonction d…

We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if $ f_1, \ldots, f_m $ are elements of the free associative algebra $…

Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of…

In this paper we study the right differentiability of a parametric infimum function over a parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to…

Green and Tao famously proved in a 2008 paper that there are arithmetic progressions of prime numbers of arbitrary lengths. Soon after, analogous statements were proved by Tao for the ring of Gaussian integers and by L\^e for the polynomial…

The class of functions from the integers to the integers computable in polynomial time has been characterized recently using discrete ordinary differential equations (ODE), also known as finite differences. In the framework of ordinary…

Let $(R,m)$ be a local Noetherian ring, let $M$ be a finitely generated $R$-module and let $(F_{\bullet},\partial_{\bullet})$ be a free resolution of $M$. We find a uniform bound $h$ such that the Artin-Rees containment $I^n F_i\cap Im \,…

We prove that, if f:R^n\to R satisfies Fr\'echet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure…

In this paper we obtain bounds for integer solutions of quadratic polynomials in two variables that represent a natural number. Also we get some results on twin prime numbers. In addition, we use linear functionals to prove some results of…

In this work we study the Artin-Mazur zeta function for piecewise monotone functions acting on a compact interval of real numbers. In the case of unimodal maps, Milnor and Thurston gave a characterization for the rationality of the…

We study the non-Archimedean Monge-Amp\`ere equation on a smooth projective variety over a discretely or trivially valued field. First, we give an example of a Green's function, associated to a divisorial valuation, which is not Q-PL (i.e.…

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose…

The goal of this paper is to study Goldbach's conjecture for rings of regular functions of affine algebraic varieties over a field. Among our main results, we define the notion of Goldbach condition for Newton polytopes, and we prove in a…

Given a number field $F$ with $\o_F$ its ring of integers. For certain $a,b$ and $\alpha$ in $\o_F$, we show that the Artin condition is the only obstruction to the local-global principle for integral solutions of equation…

Several quantities related to the Zernike circle polynomials admit an expression as an infinite integral involving the product of two or three Bessel functions. In this paper these integrals are identified and evaluated explicitly for the…

The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state…

For a rational matrix function $\Phi$ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation $\A\Phi$ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of…

Let $R$ be a finite non-commutative ring with $1\ne 0$. By a polynomial function on $R$, we mean a function $F\colon R\longrightarrow R$ induced by a polynomial $f=\sum\limits_{i=0}^{n}a_ix^i\in R[x]$ via right substitution of the variable…

We prove that functions $f:\f{2^m} \to \f{2^m}$ of the form $f(x)=x^{-1}+g(x)$ where $g$ is any non-affine polynomial are APN on at most a finite number of fields $\f{2^m}$. Furthermore we prove that when the degree of $g$ is less then 7…

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal…

We study various approximation results of solutions of equations $f(x,Y)=0$ where $f(x,Y)\in\mathbb K[[x]][Y]^r$ and $x$ and $Y$ are two sets of variables, and where some components of the solutions $y(x)\in\mathbb K[[x]]^m$ do not depend…