相关论文: Multidimensional polynomial Szemer\'{e}di theorem …

For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In 1945, Segre called for a better understanding of the large…

The (weak) Nullstellensatz over finite fields says that if $P_1,\ldots,P_m$ are $n$-variate degree-$d$ polynomials with no common zero over a finite field $\mathbb{F}$ then there are polynomials $R_1,\ldots,R_m$ such that…

We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…

Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We…

We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x = (x_1,...,x_n)$…

The principal results proved in this expository thesis are the IP polynomial Szemer\'edi theorem for nilpotent groups and the multiple term return times theorem with nilsequence weights. It also contains extensions of the convergence…

We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.

Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y),…

We show that every (possibly unbounded) convex polygon $P$ in $R^2$ with $m$ edges can be represented by inequalities $p_1 \ge 0,...,p_n \ge 0,$ where the $p_i$'s are products of at most $k$ affine functions each vanishing on an edge of $P$…

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…

When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group…

Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots,…

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…

We explore the codimension one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by…

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…

As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because…

The classical Remez inequality bounds the maximum of the absolute value of a real polynomial $P$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z\subset [-1,1]$ of positive Lebesgue measure. Extensions to…

Let~$E$ be a Hilbertian field of characteristic~$0$. R.W.K. Odoni conjectured that for every positive integer~$n$ there exists a polynomial~$f\in E[X]$ of degree~$n$ such that each iterate~$f^{\circ{k}}$ of~$f$ is irreducible and the Galois…

In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…

We present several upper bounds for the height of global residues of rational forms on an affine variety. As a consequence, we deduce upper bounds for the height of the coefficients in the Bergman-Weil trace formula. We also present upper…