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Degeneracy Loci and Polynomial Equation Solving

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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.

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Acknowledgments

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

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 , Berlin, Germany

    Bernd Bank

  2. Laboratoire d’informatique, LIX, UMR 7161 CNRS, Campus de l’École Polytechnique, 91128 , Palaiseau Cedex, France

    Marc Giusti & Grégoire Lecerf

  3. Departamento de Computación, Universidad de Buenos Aires, Ciudad Univ., Pab.I, 1428 , Buenos Aires, Argentina

    Joos Heintz

  4. CONICET, Ciudad Univ., Pab.I, 1428 , Buenos Aires, Argentina

    Joos Heintz

  5. Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39071 , Santander, Spain

    Joos Heintz

  6. Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J. M. Gutierrez 1150, B1613GSX , Los Polvorines, Buenos Aires, Argentina

    Guillermo Matera

  7. CONICET, J. M. Gutierrez 1150, B1613GSX , Los Polvorines, Buenos Aires, Argentina

    Guillermo Matera

  8. Instituto Matemático Luis Santaló, CONICET, Buenos Aires, 1428, Argentina

    Pablo Solernó

  9. Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina

    Pablo Solernó

Authors
  1. Bernd Bank
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  2. Marc Giusti
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  3. Joos Heintz
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  4. Grégoire Lecerf
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  5. Guillermo Matera
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  6. Pablo Solernó
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Corresponding author

Correspondence to Joos Heintz.

Additional information

Communicated by Teresa Krick and James Renegar.

Dedicated to Mike Shub on the occasion of his 70th birthday.

Research partially supported by the following Argentinian, French and Spanish grants:

CONICET Res 4541-12, PIP 11220090100421 CONICET, UBACYT 20020100100945 and 20020110100063, PICT-2010-0525, Digiteo DIM 2009-36HD “MaGiX” grant of the Région Ile-de-France, ANR-2010-BLAN-0109-04 “LEDA”, MTM2010-16051.

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Bank, B., Giusti, M., Heintz, J. et al. Degeneracy Loci and Polynomial Equation Solving. Found Comput Math 15, 159–184 (2015). https://doi.org/10.1007/s10208-014-9214-z

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