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Computing Complex Dimension Faster and Deterministically

J. Maurice Rojas

TL;DR

Addressing the problem of computing the complex dimension $\dim Z_F$ of the zero set of a sparse polynomial system $F$, the paper introduces a completely deterministic algorithm based on toric resultants that reduces multivariate solving to matrix and univariate arithmetic, achieving a bound of $\mathcal{O}(n^{2.312} 11^{n} k V_F^{7.376})$ and accompanying height bounds. It develops sharp rational univariate representation (RUR) bounds, provides explicit coefficient-height controls, and constructs an explicit univariate witness $h_F$ along with $u_F$ for $Z_F$, with complexity matching the main dimension bound. Furthermore, under GRH, it refines Koiran’s hypersurface-emptiness test by giving explicit density-based constants $a_F$ and $A_F$ that certify feasibility through modular reductions, thereby yielding a sharper conditional algorithm. Together, the results advance deterministic, sparsity-aware dimension computation over $\mathbb{C}$ and offer concrete, scalable tools for symbolic solving and decision problems in algebraic geometry, while clarifying the role of polytopal volume $V_F$ in complexity.

Abstract

We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper quantitative estimates on rational univariate representations of roots of polynomial systems. As a corollary of the latter bounds, we considerably improve a recent algorithm of Koiran for deciding the emptiness of a hypersurface intersection over the complex numbers, given the truth of the Generalized Riemann Hypothesis (GRH).

Computing Complex Dimension Faster and Deterministically

TL;DR

Addressing the problem of computing the complex dimension of the zero set of a sparse polynomial system , the paper introduces a completely deterministic algorithm based on toric resultants that reduces multivariate solving to matrix and univariate arithmetic, achieving a bound of and accompanying height bounds. It develops sharp rational univariate representation (RUR) bounds, provides explicit coefficient-height controls, and constructs an explicit univariate witness along with for , with complexity matching the main dimension bound. Furthermore, under GRH, it refines Koiran’s hypersurface-emptiness test by giving explicit density-based constants and that certify feasibility through modular reductions, thereby yielding a sharper conditional algorithm. Together, the results advance deterministic, sparsity-aware dimension computation over and offer concrete, scalable tools for symbolic solving and decision problems in algebraic geometry, while clarifying the role of polytopal volume in complexity.

Abstract

We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper quantitative estimates on rational univariate representations of roots of polynomial systems. As a corollary of the latter bounds, we considerably improve a recent algorithm of Koiran for deciding the emptiness of a hypersurface intersection over the complex numbers, given the truth of the Generalized Riemann Hypothesis (GRH).

Paper Structure

This paper contains 10 sections, 16 theorems, 3 equations.

Table of Contents

  1. Introduction and Main Results
  2. Related Results Over $\mathbb{C}$
  3. A Sparse $\pmb{3\times 3}$ Example
  4. Proofs of Our Results Over $\mathbb{C}$: Theorems \ref{['main:complex']}, \ref{['main:height']}, \ref{['main:size']}, \ref{['main:unired']}, and \ref{['main:koi']}
  5. The Proof of Theorem \ref{['main:complex']}
  6. The Proof of Theorem \ref{['main:size']}
  7. The Proof of Theorem \ref{['main:unired']}
  8. The Proof of Theorem \ref{['main:koi']}
  9. Acknowledgements
  10. Appendix: Background on Toric Resultants and The Proof of Theorem \ref{['main:height']}

Key Result

Theorem 1

Let $D$ the maximum of the total degrees of $f_1,\ldots,f_m$. Also let $\mathbf{O}$ be the origin, and $e_1,\ldots,e_n$ the standard basis vectors, in $\mathbb{R}^n$. Normalize $n$-dimensional volume $\mathrm{Vol}_n(\cdot)$ so that the standard $n$-simplex (with vertices $\mathbf{O},e_1,\ldots,e_n$)

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Remark 4
  • ...and 17 more