Related papers: Generalized "second Ritt theorem" and explicit sol…
In these short notes, we will show the following. Let F_q be a finite field and let E/\F_q be an elliptic curve. Let S_r be the rth summation/Semaev polynomial for E. Under an assumption, we show that it is NP-complete to check if S_r…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…
This article finds the answer to the question: for any problem from which a non-deterministic algorithm can be derived which verifies whether an answer is correct or not in polynomial time (complexity class NP), is it possible to create an…
We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem…
This paper investigates defining equations for secant varieties of the variety of reducible polynomials, which geometrically encode the notions of strength and slice rank of homogeneous polynomials. We present three main results. First, we…
This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to…
Associated to each random variable $Y$ having a finite moment generating function, we introduce a different generalization of the Stirling numbers of the second kind. Some characterizations and specific examples of such generalized numbers…
We provide a necessary and sufficient condition for the representability of a function as the classical multidimensional Laplace transform, when the support of the representing measure is contained in some generalized semi-algebraic set.…
Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…
This paper presents new six solutions for sixth degree polynomial equation in general forms basing on new theorems, where the possibility to calculate the six roots of any sixth degree equation nearly simultaneously. The proposed roots for…
We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral…
Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the…
The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of…
In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
We consider the problem of decomposing a regular non-negative function as a sum of squares of functions which preserve some form of regularity. In the same way as decomposing non-negative polynomials as sum of squares of polynomials allows…
We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P in R^d is a rational function. Its denominator is the product of linear…
This paper continues the functional approach to the P-versus-NP problem, begun in [1]. Here we focus on the monoid RM_2^P of right-ideal morphisms of the free monoid, that have polynomial input balance and polynomial time-complexity. We…