Degeneracy Loci and Polynomial Equation Solving
| Autor: | Pablo Solernó, Bernd Bank, Joos Heintz, Marc Giusti, Grégoire Lecerf, Guillermo Matera |
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| Rok vydání: | 2014 |
| Předmět: |
FOS: Computer and information sciences
Computer Science - Symbolic Computation Endomorphism Matemáticas Dimension (graph theory) Vector bundle Symbolic Computation (cs.SC) Rank (differential topology) Equidimensional Matemática Pura Combinatorics Mathematics - Algebraic Geometry FOS: Mathematics Algebraic Geometry (math.AG) Mathematics Polynomial (hyperelastic model) Discrete mathematics Degree (graph theory) DEGREE OF VARIETIES Applied Mathematics PSEUDO-POLYNOMIAL COMPLEXITY POLYNOMIAL EQUATION SOLVING Computational Mathematics Computational Theory and Mathematics Affine space 14Q20 (Primary) 14M10 14M12 14P05 68W30 (Secondary) DEGENERACY LOCUS CIENCIAS NATURALES Y EXACTAS Analysis |
| Zdroj: | Foundations of Computational Mathematics. 15:159-184 |
| ISSN: | 1615-3383 1615-3375 |
| DOI: | 10.1007/s10208-014-9214-z |
| Popis: | Let V be a smooth equidimensional quasi-affine variety of dimension r over the complex numbers $C$ and let $F$ be a $(p\times s)$-matrix of coordinate functions of $C[V]$, where $s\ge p+r$. The pair $(V,F)$ determines a vector bundle $E$ of rank $s-p$ over $W:=\{x\in V:\mathrm{rk} F(x)=p\}$. 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 which we are going to design and which 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. 24 pages, accepted for publication in Found. Comput. Math |
| Databáze: | OpenAIRE |
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