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

Patil, DP and Verma, JK (2022) Rational Points and Trace Forms on a Finite Algebra over a Real Closed Field. [Book Chapter]

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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.

Item Type: Book Chapter
Publication: Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of his 75th Birthday
Publisher: Springer International Publishing
Additional Information: The copyright of this article belongs to Springer International Publishing.
Keywords: Finite K-algebra; Hermitian forms; Quadratic forms; Real closed fields; Signature; Sylvester’s law of inertia; Trace forms; Type
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 14 Jun 2023 09:46
Last Modified: 14 Jun 2023 09:46
URI: https://eprints.iisc.ac.in/id/eprint/81882

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