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Foundations of Computational Mathematics

, Volume 15, Issue 1, pp 159–184 | Cite as

Degeneracy Loci and Polynomial Equation Solving

  • Bernd Bank
  • Marc Giusti
  • Joos Heintz
  • Grégoire Lecerf
  • Guillermo Matera
  • Pablo Solernó
Article

Abstract

Let \(V\) be a smooth, equidimensional, quasi-affine variety of dimension \(r\) over \(\mathbb {C}\), and let \(F\) be a \((p\times s)\) matrix of coordinate functions of \(\mathbb {C}[V]\), where \(s\ge p+r\). The pair \((V,F)\) determines a vector bundle \(E\) of rank \(s-p\) over \(W:=\{x\in V \mid \mathrm{rk }F(x)=p\}\). We associate with \((V,F)\) a descending chain of degeneracy loci of \(E\) (the generic polar varieties of \(V\) represent a typical example of this situation). The maximal degree of these degeneracy loci constitutes the essential ingredient for the uniform, bounded-error probabilistic pseudo-polynomial-time algorithm that we will design and that solves a series of computational elimination problems that can be formulated in this framework. We describe applications to polynomial equation solving over the reals and to the computation of a generic fiber of a dominant endomorphism of an affine space.

Keywords

Polynomial equation solving Pseudo-polynomial complexity Degeneracy locus Degree of varieties 

Mathematics Subject Classification

14M10 14M12 14Q20 14P05 68W30 

Notes

Acknowledgments

The authors wish to thank Antonio Campillo (Valladolid, Spain) for stimulating conversations on the subject of this paper.

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Copyright information

© SFoCM 2014

Authors and Affiliations

  • Bernd Bank
    • 1
  • Marc Giusti
    • 2
  • Joos Heintz
    • 3
    • 4
    • 5
  • Grégoire Lecerf
    • 2
  • Guillermo Matera
    • 6
    • 7
  • Pablo Solernó
    • 8
    • 9
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Laboratoire d’informatiqueLIX, UMR 7161 CNRSPalaiseau CedexFrance
  3. 3.Departamento de ComputaciónUniversidad de Buenos AiresBuenos AiresArgentina
  4. 4.CONICETBuenos AiresArgentina
  5. 5.Departamento de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain
  6. 6.Instituto del Desarrollo HumanoUniversidad Nacional de General SarmientoLos Polvorines, Buenos AiresArgentina
  7. 7.CONICETLos Polvorines, Buenos AiresArgentina
  8. 8.Instituto Matemático Luis SantalóCONICETBuenos AiresArgentina
  9. 9.Departamento de Matemáticas, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina

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