The expansion of real forms on the simplex and applications

TL;DR

This paper investigates the relationship between the geometry of simplices in the standard simplex and the signs of coefficients of forms evaluated on these simplices, providing criteria for real zeros of algebraic systems.

Contribution

It proves a new geometric criterion linking simplex size and point inclusion to the sign of polynomial coefficients, and applies this to determine real zeros of algebraic systems.

Findings

Small-diameter simplices containing a point imply sign conditions on polynomial coefficients.

A necessary and sufficient condition for real zeros of algebraic systems on the simplex.

Application to systems with integral coefficients.

Abstract

If n points B_1,---,B_n$ in the standard simplex \Delta_n are affinely independent, then they can span an (n-1)-simplex denoted by \Lambda=Con(B_1,---,B_n). Here \Lambda corresponds to an n*n matrix [\Lambda] whose columns are B_1,---,B_n. In this paper, we firstly proved that if \Lambda of diameter sufficiently small contains a point $P$, and f(P)>0 (<0) for a form f in R[X], then the coefficients of f([\Lambda] X) are all positive (negative). Next, as an application of this result, a necessary and sufficient condition for determining the real zeros on \Delta_n of a system of homogeneous algebraic equations with integral coefficients is established.