Rational Points and Trace Forms on a Finite Algebra over a Real Closed Field

Dilip P. Patil () and J. K. Verma ()
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Dilip P. Patil: Indian Institute of Science, Department of Mathematics
J. K. Verma: Indian Institute of Technology Bombay, Mathematics Department

A chapter in Commutative Algebra, 2021, pp 669-687 from Springer

Abstract: Abstract The main goal of this article is to provide a proof of the Pederson-Roy-Szpirglas theorem about counting the number of 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 of the number of K-rational points of a finite 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.

Keywords: Real closed fields; Finite K-algebra; Hermitian forms; Quadratic forms; Sylvester’s law of inertia; Type; Signature; Trace forms; Primary 13-02; 13B22; 13F30; 13H15; 14C17 (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-89694-2_22

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DOI: 10.1007/978-3-030-89694-2_22

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