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

In: Commutative Algebra

Author

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  • Dilip P. Patil

    (Indian Institute of Science, Department of Mathematics)

  • J. K. Verma

    (Indian Institute of Technology Bombay, Mathematics Department)

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.

Suggested Citation

  • Dilip P. Patil & J. K. Verma, 2021. "Rational Points and Trace Forms on a Finite Algebra over a Real Closed Field," Springer Books, in: Irena Peeva (ed.), Commutative Algebra, pages 669-687, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-89694-2_22
    DOI: 10.1007/978-3-030-89694-2_22
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