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Future Vision and Trends on Shapes, Geometry and Algebra pp 55–71Cite as

Sparse Discriminants and Applications

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 84))

Abstract

Polynomial algebra offers a standard, powerful, and robust approach to handle several important problems in geometric modelling and other areas. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the existence of multiple (or degenerate) roots. We describe discriminants in a general context, and relate them to an equally useful object, namely the resultant of an overconstrained polynomial system. We discuss several relevant applications in geometric modelling so as to motivate the use of such algebraic tools in further geometric problems. We focus on exploiting the sparseness of polynomials via the theory of Newton polytopes and sparse elimination.

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Acknowledgments

We thank Alicia Dickenstein for useful discussions. I. Z. Emiris was partially supported by Marie-Curie Initial Training Network “SAGA” (ShApes, Geometry, Algebra), FP7-PEOPLE contract PITN-GA-2008-214584. A. Karasoulou’s research leading to these results has received funding from the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program “ARISTEIA”, project “ESPRESSO: Exploiting Structure in Polynomial Equation and System Solving in Physical Modeling".

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Authors and Affiliations

  1. Department of Informatics and Telecommunications, University of Athens, Athens, Greece

    Ioannis Z. Emiris & Anna Karasoulou

Authors
  1. Ioannis Z. Emiris
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  2. Anna Karasoulou
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Correspondence to Ioannis Z. Emiris .

Editor information

Editors and Affiliations

  1. Fondazione Graphitech, Povo, Trento, Italy

    Raffaele De Amicis

  2. Fondazione Graphitech, Povo, Trento, Italy

    Giuseppe Conti

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Emiris, I.Z., Karasoulou, A. (2014). Sparse Discriminants and Applications. In: De Amicis, R., Conti, G. (eds) Future Vision and Trends on Shapes, Geometry and Algebra. Springer Proceedings in Mathematics & Statistics, vol 84. Springer, London. https://doi.org/10.1007/978-1-4471-6461-6_4

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