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Chebyshev Varieties

Zaïneb Bel-Afia, Chiara Meroni, Simon Telen

TL;DR

The paper introduces Chebyshev varieties as Chebyshev-analogue counterparts to toric varieties for sparse polynomial systems, focusing on when monomials are replaced by Chebyshev polynomials or cosines. It develops three main families—plane/space Chebyshev curves, tensor-product Chebyshev varieties, and cosine Chebyshev varieties—and establishes dimension and degree results, singular loci descriptions, and defining equations under natural arithmetic conditions. It then presents numerical algorithms for solving Chebyshev-based systems, including a linear-algebra approach for tensor-product equations and homotopy continuation methods for cosine-based equations, along with applications to real-root distribution, root-system generalizations, and potential future directions. Overall, the work provides a rigorous geometric framework and practical tools for Chebyshev-based sparse solving, offering parallels to toric geometry in the monomial setting and enabling efficient numerical root finding in multiple variables.

Abstract

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. We determine the dimension, degree, singular locus and defining equations of these varieties. We explain how they play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We present numerical root finding algorithms that exploit our results.

Chebyshev Varieties

TL;DR

The paper introduces Chebyshev varieties as Chebyshev-analogue counterparts to toric varieties for sparse polynomial systems, focusing on when monomials are replaced by Chebyshev polynomials or cosines. It develops three main families—plane/space Chebyshev curves, tensor-product Chebyshev varieties, and cosine Chebyshev varieties—and establishes dimension and degree results, singular loci descriptions, and defining equations under natural arithmetic conditions. It then presents numerical algorithms for solving Chebyshev-based systems, including a linear-algebra approach for tensor-product equations and homotopy continuation methods for cosine-based equations, along with applications to real-root distribution, root-system generalizations, and potential future directions. Overall, the work provides a rigorous geometric framework and practical tools for Chebyshev-based sparse solving, offering parallels to toric geometry in the monomial setting and enabling efficient numerical root finding in multiple variables.

Abstract

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. We determine the dimension, degree, singular locus and defining equations of these varieties. We explain how they play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We present numerical root finding algorithms that exploit our results.
Paper Structure (14 sections, 20 theorems, 86 equations, 11 figures)

This paper contains 14 sections, 20 theorems, 86 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Contributions and outline.
  4. Toric varieties and sparse root finding
  5. Chebyshev curves
  6. Plane curves
  7. Space curves
  8. Tensor product Chebyshev polynomials
  9. Cosines of linear forms
  10. Applications and future directions
  11. Real roots
  12. Solving tensor product Chebyshev equations
  13. Solving cosine equations
  14. Root systems

Key Result

Theorem 2.1

The dimension of the affine toric variety ${\cal Y}_A$ equals the rank of the matrix $A$, which equals the dimension of the polytope $P_A = {\rm conv}(A \cup 0)$. If ${\rm rank}(A) = m$, we have The ideal of ${\cal Y}_A$ is generated by the binomials $x^u - x^v$, for $u, v \in \mathbb{N}^n$ and $u-v \in \ker_{\mathbb{Z}} A$.

Figures (11)

  • Figure 1: Three surfaces obtained from the matrix $A = \left [ 112213 \right ]$.
  • Figure 2: The number of intersection points of ${\cal Y}_A$ is $2!$ times the volume of $P_A$.
  • Figure 3: Chebyshev curves intersected by lines passing through the origin.
  • Figure 4: Chebyshev $U$-curves with different singularities.
  • Figure 5: A smooth (left) and a singular (right) Chebyshev space curve.
  • ...and 6 more figures

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 29 more