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Some Lower Bounds on the Reach of an Algebraic Variety

Chris La Valle, Josué Tonelli-Cueto

TL;DR

This work investigates lower bounds on the reach $\varrho(Z)$ of smooth real algebraic varieties, framing reach as the positive-dimensional analogue of zero-set separation. It develops three complementary strands: (i) direct reach bounds via Kantorovich and Smale theory that tie geometric reach to condition numbers and regularity measures, (ii) Federer's estimators to connect local geometry with global reach, and (iii) probabilistic bounds for random polynomial models (continuous and discrete/bit-sized) that quantify how reach behaves under randomness. The results yield explicit worst-case and probabilistic bounds, including bounds that scale with degree, bit-size, and distance to infinity, and demonstrate how reach can be controlled by conditioning and derivative information. These bounds underpin certifiable geometric inference for algebraic sets and enhance numerical robustness in solving and sampling real varieties.

Abstract

Separation bounds are a fundamental measure of the complexity of solving a zero-dimensional system as it measures how difficult it is to separate its zeroes. In the positive dimensional case, the notion of reach takes its place. In this paper, we provide bounds on the reach of a smooth algebraic variety in terms of several invariants of interest: the condition number, Smale's $γ$ and the bit-size. We also provide probabilistic bounds for random algebraic varieties under some general assumptions.

Some Lower Bounds on the Reach of an Algebraic Variety

TL;DR

This work investigates lower bounds on the reach of smooth real algebraic varieties, framing reach as the positive-dimensional analogue of zero-set separation. It develops three complementary strands: (i) direct reach bounds via Kantorovich and Smale theory that tie geometric reach to condition numbers and regularity measures, (ii) Federer's estimators to connect local geometry with global reach, and (iii) probabilistic bounds for random polynomial models (continuous and discrete/bit-sized) that quantify how reach behaves under randomness. The results yield explicit worst-case and probabilistic bounds, including bounds that scale with degree, bit-size, and distance to infinity, and demonstrate how reach can be controlled by conditioning and derivative information. These bounds underpin certifiable geometric inference for algebraic sets and enhance numerical robustness in solving and sampling real varieties.

Abstract

Separation bounds are a fundamental measure of the complexity of solving a zero-dimensional system as it measures how difficult it is to separate its zeroes. In the positive dimensional case, the notion of reach takes its place. In this paper, we provide bounds on the reach of a smooth algebraic variety in terms of several invariants of interest: the condition number, Smale's and the bit-size. We also provide probabilistic bounds for random algebraic varieties under some general assumptions.
Paper Structure (19 sections, 25 theorems, 42 equations)

This paper contains 19 sections, 25 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Federer's Estimators of the Reach
  4. 1-Norm on Polynomials
  5. Kantorovich, Smale and the Reach
  6. Kantorovich's Theory and the Reach
  7. Smale's alpha-theory and the Reach
  8. Cubic Condition Number and Reach
  9. Properties of the Condition Number
  10. Condition-based Bound on Reach
  11. Random Bounds for the Reach: the Continuous Case
  12. Zintzo random polynomial tuples
  13. Condition of a zintzo random polynomial
  14. Bounds on the Reach: Proof of Theorem \ref{['theo:probcaseboundreach_contunif']}
  15. Integer Bounds for the Reach
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $R\in\mathbb{N}$ and $f\in\mathcal{P}_{n,\boldsymbol{d}}[q]$ be a tuple of integer polynomials with coefficients of bit-size at most $\tau$. Then $\mathop{\mathrm{\varrho}}\nolimits_R(\mathcal{Z}(f))=0$ or $\log 1/\mathop{\mathrm{\varrho}}\nolimits_R(\mathcal{Z}(f))$ is (upper) bounded by

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Corollary 4.4
  • Definition 5.1
  • ...and 29 more