Related papers: Rational points and prime values of polynomials in…

We consider equations of the form $a_{1}x_{1}^{k}+...+a_{s}x_{s}^{k}$ and when they have solutions in the primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove…

For a field $E$ of characteristic different from $2$ and cohomological $2$-dimension one, quadratic forms over the rational function field $E(X)$ are studied. A characterisation in terms of polynomials in $E[X]$ is obtained for having that…

There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to…

This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining…

Let $P$ be a bounded convex subset of $\mathbb R^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots,X_n)$ vanishing on $P\cap\mathbb Z^n$ by $r_P$ and denote the smallest number $u\geq0$ such that every function…

In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of…

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $[-1,1]$ if they are bounded by $1$ on a subset of $[-1,1]$ of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev…

We discuss as a fundamental characteristic of orthogonal polynomials like the existence of a Lie algebra behind them, can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we put…

We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial…

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and…

We derive, via the Hardy-Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of non-singular local solubility. Our polynomials $F({\mathbf…

We investigate the structure of polynomials of degree four in many variables over a fixed prime field $\mathbb{F}=\mathbb{F}_{p}$. In 2007, Green and Tao proved that if a polynomial $f:\mathbb{F}^{n}\rightarrow\mathbb{F}$ is poorly…

Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-\alpha x+1$ for $\alpha=3,4,5,\dots$.Some of the underlying identities suggest an identity involving…

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of union of edges. These polynomials arise naturally to model job-occupancy in…

Fix $k,s,n\in \mathbb N$, and consider non-zero integers $c_1,\ldots ,c_s$, not all of the same sign. Provided that $s\ge k(k+1)$, we establish a Hasse principle for the existence of lines having integral coordinates lying on the affine…

In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of…

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact,…

We show that Schur classes of ample vector bundles on smooth projective varieties satisfy Hodge-Riemann relations on $H^{p,q}$ under the assumption that $H^{p-2,q-2}$ vanishes. More generally, we study Hodge-Riemann polynomials, which are…

Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring…

The problem of characterizing a real polynomial $f$ as a sum of squares of polynomials on a real algebraic variety $V$ dates back to the pioneering work of Hilbert in [Mathematische Annalen 32.3 (1888): 342-350]. In this paper, we…