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A semicontinuous relaxation of Saito's criterion and freeness as angular minimization

Tomás S. R. Silva

Abstract

We introduce a nonnegative functional on the space of line arrangements in $\mathbb{P}^2$ that vanishes precisely on free arrangements, obtained as a semicontinuous relaxation of Saito's criterion for freeness. Given an arrangement $\mathcal{A}$ of $n$ lines with candidate exponents $(d_1, d_2)$, we parameterize the spaces of logarithmic derivations of degrees $d_1$ and $d_2$ via the null spaces of the associated derivation matrices and express the Saito determinant as a bilinear map into the space of degree $n$ polynomials. The functional then admits a natural geometric interpretation: it measures the squared sine of the angle between the image of this bilinear map and the direction of the defining polynomial $Q(\mathcal{A})$ in coefficient space, and equals zero if and only if its image contains the line spanned by $Q(\mathcal{A})$. This provides a computable measure of how far a given arrangement is from admitting a free basis of logarithmic derivations of the expected degrees. Using this functional as a reward signal, we develop a sequential construction procedure in which lines are added one at a time so as to minimize the angular distance to freeness, implemented via reinforcement learning with an adaptive curriculum over arrangement sizes and exponent types. Our results suggest that semicontinuous relaxation techniques, grounded in the geometry of polynomial coefficient spaces, offer a viable approach to the computational exploration of freeness in the theory of line arrangements.

A semicontinuous relaxation of Saito's criterion and freeness as angular minimization

Abstract

We introduce a nonnegative functional on the space of line arrangements in that vanishes precisely on free arrangements, obtained as a semicontinuous relaxation of Saito's criterion for freeness. Given an arrangement of lines with candidate exponents , we parameterize the spaces of logarithmic derivations of degrees and via the null spaces of the associated derivation matrices and express the Saito determinant as a bilinear map into the space of degree polynomials. The functional then admits a natural geometric interpretation: it measures the squared sine of the angle between the image of this bilinear map and the direction of the defining polynomial in coefficient space, and equals zero if and only if its image contains the line spanned by . This provides a computable measure of how far a given arrangement is from admitting a free basis of logarithmic derivations of the expected degrees. Using this functional as a reward signal, we develop a sequential construction procedure in which lines are added one at a time so as to minimize the angular distance to freeness, implemented via reinforcement learning with an adaptive curriculum over arrangement sizes and exponent types. Our results suggest that semicontinuous relaxation techniques, grounded in the geometry of polynomial coefficient spaces, offer a viable approach to the computational exploration of freeness in the theory of line arrangements.

Paper Structure

This paper contains 29 sections, 4 theorems, 41 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Line arrangements in P2
  3. Characteristic polynomial and Betti numbers
  4. Logarithmic derivations
  5. Free arrangements and Saito's criterion
  6. Terao's factorization and numerical constraints
  7. Relevance for the computational framework
  8. Background on machine learning
  9. Reinforcement learning
  10. The Saito functional and the search framework
  11. Sequential construction of arrangements
  12. A semicontinuous relaxation of Saito's criterion
  13. Derivation matrices
  14. Null space parameterization
  15. The bilinear determinant map
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\theta_0, \theta_1, \theta_2 \in D(\mathcal{A})$ be homogeneous derivations, and write $\theta_i = f_{i1}\,\partial_x + f_{i2}\,\partial_y + f_{i3}\,\partial_z$. Form the coefficient matrix Then $\mathcal{A}$ is free with basis $\{\theta_0, \theta_1, \theta_2\}$ if and only if for some $c \in \mathbb{C}^\times$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (12)

  • Theorem 1: Saito's criterion Saito1980
  • Theorem 2: Terao's factorization theorem Terao1981
  • Remark 1
  • Remark 2
  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2: Upper semicontinuity
  • proof
  • Conjecture 1: Terao's conjecture, functional formulation
  • ...and 2 more