Related papers: Degrees of Second and Higher-Order Polynomials

This paper investigates the p-adic valuation trees of degree-2 and degree-3 polynomials in two variables over any prime p, building upon prior research outlined in [14].

We consider semiclassical orthogonal polynomials on the unit circle associated with a weight function that satisfy a Pearson-type differential equation involving two polynomials of degree at most three. Structure relations and difference…

In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual…

The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of…

These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones.…

Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the…

The p-adic valuation of a polynomial can be given by its valuation tree. This work describes the 2-adic valuation tree of the general degree 2 polynomial in 2 variables.

We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…

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…

A quasi-ordinary polynomial is a monic polynomial with coefficients in the power series ring such that its discriminant equals a monomial up to unit. In this paper we study higher derivatives of quasi-ordinary polynomials, also called…

A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal…

The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…

Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets…

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…

In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent…

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on…

In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it…