The expansion of real forms on the simplex and applications

Abstract

If n points B_1,---,B_n$inthestandardsimplex{\Delta }_{n}areaffinelyindependent,thentheycanspanan(n-1)-simplexdenotedby\Lambda =Con(B_1,---,B_n).Here\Lambda correspondstoann\ast nmatrix[\Lambda ]whosecolumnsare{B}_1,---,B_n.Inthispaper,wefirstlyprovedthatif\Lambda ofdiametersufficientlysmallcontainsapoint$P$, and f(P)>0 (<0) for a form f in R[X], then the coefficients of f([Λ] X) are all positive (negative). Next, as an application of this result, a necessary and sufficient condition for determining the real zeros on Δ_n of a system of homogeneous algebraic equations with integral coefficients is established.

10 pages, 1 figures