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K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.

Polynomial set-valued functions

Joanna Szczawińska (1996)

Annales Polonici Mathematici

The aim of this paper is to give a necessary and sufficient condition for a set-valued function to be a polynomial s.v. function of order at most 2.

Polynomial two-point expansions.

A. Clausing (1988)

Aequationes mathematicae

Polynomials and convex sequence inequalities.

Mercer, A.McD. (2005)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Polynomials and degrees of maps in real normed algebras

Takis Sakkalis (2020)

Communications in Mathematics

Let $𝒜$ be the algebra of quaternions $ℍ$ or octonions $𝕆$ . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial $f (t) \in 𝒜 [t]$ has a root in $𝒜$ . As a consequence, the Jacobian determinant $| J (f) |$ is always non-negative in $𝒜$ . Moreover, using the idea of the topological degree we show that a regular polynomial $g (t)$ over $𝒜$ has also a root in $𝒜$ . Finally, utilizing multiplication ( $*$ ) in $𝒜$ , we prove various results on the topological degree of products...

Polynomials, sign patterns and Descartes' rule of signs

Vladimir Petrov Kostov (2019)

Mathematica Bohemica

By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ( $c + p = d$ ) has $pos \leq c$ positive and $\neg \leq p$ negative roots, where $pos \equiv c (mod 2)$ and $\neg \equiv p (mod 2)$ . For $1 \leq d \leq 3$ , for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(pos, neg)$ satisfying these conditions there exists a polynomial $P$ with exactly $pos$ positive and exactly $\neg$ negative roots (all of them simple). For $d \geq 4$ this is not...