Rational points and generalized trace forms on a finite algebra over a real closed field

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].