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

## TL;DR

This paper provides a new proof of the Pederson-Roy-Szpirglas theorem by connecting counting real solutions of polynomial equations to invariants of trace forms on finite algebras over real closed fields.

## Contribution

It offers a novel proof of a counting theorem using linear algebra and algebraic invariants, linking algebraic geometry and form theory.

## Key findings

- Proof of Pederson-Roy-Szpirglas theorem using trace form signatures
- Establishes equality between rational points count and trace form signature
- Connects algebraic invariants with real zero counting

## Abstract

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].