  The main goal of this article is to provide a proof of the
Pederson-Roy-Szpirglas theorem about counting common real zeros of real
polynomial equations by using basic results from Linear algebra and Commutative
algebra. The main tools are symmetric bilinear forms, Hermitian forms, trace
forms, and their invariants such as rank, types, and signatures. Further, we
use the equality (proved in [3]) of the number of K-rational points of a
zero-dimensional affine algebraic set over a real closed field $K$ with the
signature of the trace form of its coordinate ring to prove the
Pederson-Roy-Szpirglas theorem, see [16].
