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Sparse systems and algorithmic equidimensional decomposition

Maria Isabel Herrero, Gabriela Jeronimo, Juan Sabia

TL;DR

A new probabilistic algorithm is presented that characterizes the equidimensional components of the affine algebraic variety defined by an arbitrary sparse polynomial system with prescribed supports, namely a finite set obtained by intersecting the component with a generic linear variety of complementary dimension.

Abstract

We present a new probabilistic algorithm that characterizes the equidimensional components of the affine algebraic variety defined by an arbitrary sparse polynomial system with prescribed supports. For each equidimensional component, the algorithm computes a witness set, namely a finite set obtained by intersecting the component with a generic linear variety of complementary dimension. The complexity of the algorithm is polynomial in combinatorial invariants associated to the supports of the polynomials involved.

Sparse systems and algorithmic equidimensional decomposition

TL;DR

A new probabilistic algorithm is presented that characterizes the equidimensional components of the affine algebraic variety defined by an arbitrary sparse polynomial system with prescribed supports, namely a finite set obtained by intersecting the component with a generic linear variety of complementary dimension.

Abstract

We present a new probabilistic algorithm that characterizes the equidimensional components of the affine algebraic variety defined by an arbitrary sparse polynomial system with prescribed supports. For each equidimensional component, the algorithm computes a witness set, namely a finite set obtained by intersecting the component with a generic linear variety of complementary dimension. The complexity of the algorithm is polynomial in combinatorial invariants associated to the supports of the polynomials involved.
Paper Structure (13 sections, 10 theorems, 27 equations)

This paper contains 13 sections, 10 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sparse polynomial systems
  4. Affine varieties, equidimensional decomposition and witness sets
  5. Complexity estimates
  6. Theoretical results
  7. A single point
  8. A finite set of points
  9. Algorithms
  10. Geometric resolutions
  11. Algorithmic deformation to classify points
  12. Computation of witness point sets
  13. Conclusion

Key Result

Lemma 1

With the previous notation, we have that $\mathcal{Z}_h \subset V_{h+k-1}\times \mathbb{A}^1$ for $h=1, \dots, n-k$, and $\mathcal{Z}_h= \emptyset$ otherwise.

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Lemma 4
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • Proposition 8
  • Remark 9
  • Proposition 10
  • ...and 2 more