On the factorization of linear combinations of polynomials

TL;DR

This paper investigates conditions under which all linear combinations of certain second-degree polynomials are factorable, providing characterizations and minimal requirements for such factorizability in both bivariate and trivariate cases.

Contribution

It introduces new characterizations of polynomials with universally factorizable linear combinations and determines the minimal number of such combinations needed.

Findings

Characterization of polynomials with all linear combinations factorizable

Determination of minimal number of factorizable combinations for universality

Extension of results from bivariate to trivariate polynomials

Abstract

In this paper we consider linear combinations of two trivariate homogeneous polynomials of second degree. We formulate and solve two problems: i) Characterization of polynomials for which all linear combinations are factorizable. ii) How many linear factorizable combinations are required for all linear combinations to be factorizable. Next, the solutions of analog problems for bivariate polynomials of second degree are derived.