The expansion of real forms on the simplex and applications

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.