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 $\mathbb{P}^2$ 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.

• The paper introduces a novel angular minimization functional that relaxes Saito's criterion and quantifies deviation from freeness.
• It employs an alternating least squares optimization within a reinforcement learning framework to construct free line arrangements.
• Numerical results validate the approach on moderate-sized line configurations, offering explicit arrangements and insights into Terao's conjecture.

Semicontinuous Relaxation of Saito's Criterion and Freeness as Angular Minimization

Introduction

This paper presents a novel approach to the computational construction and analysis of free line arrangements in the complex projective plane $\mathbb{P}^2$. Building on foundational concepts in the theory of hyperplane arrangements, the work proposes a semicontinuous, numerically tractable relaxation of Saito's criterion for arrangement freeness. The key technical innovation is the introduction of a functional that quantitatively measures the angular deviation from freeness in the space of coefficient vectors of homogeneous polynomials, thereby embedding a discrete algebraic property into a continuous optimization landscape. This functional is then employed as a mathematically principled reward signal within a reinforcement learning (RL) framework, allowing for the automated, sequential synthesis of free arrangements.

Freeness and Saito's Criterion: Background

The freeness of a line arrangement $\mathcal{A}$ in $\mathbb{P}^2$ is characterized in terms of the splitting of the $S$-module of logarithmic derivations $D(\mathcal{A})$. Saito's criterion provides an explicit condition: $\mathcal{A}$ is free with exponents $(d_1, d_2)$ if and only if there exist homogeneous derivations $\theta_1,\theta_2 \in D(\mathcal{A})$ of degrees $d_1$ and $d_2$ such that the determinant of the coefficient matrix formed with $\mathcal{A}$0 (the Euler derivation) satisfies $\mathcal{A}$1 for some $\mathcal{A}$2, with $\mathcal{A}$3 the defining polynomial. While this determinant condition is computationally checkable, verifying it exhaustively or constructing arrangements meeting it a priori has been intractable for nontrivial $\mathcal{A}$4 due to the large parameter space and the scarcity of free realizations.

Semicontinuous Relaxation via Angular Minimization

The main methodological advance is to reinterpret Saito's criterion as a geometric incidence in coefficient space. The authors parameterize the space of logarithmic derivations of fixed degrees as null spaces of linear derivation matrices. The determinant construction in Saito's criterion is then realized as a bilinear map $\mathcal{A}$5, where $\mathcal{A}$6 are the null space dimensions and $\mathcal{A}$7 is the dimension of degree-$\mathcal{A}$8 forms.

The core functional is defined as: $\mathcal{A}$9 which computes the squared sine of the minimal angle between the image of $\mathbb{P}^2$0 and the coefficient vector of $\mathbb{P}^2$1. The functional is nonnegative, vanishing if and only if freeness holds (i.e., the Saito determinant can be made proportional to $\mathbb{P}^2$2), and varying semicontinuously with the arrangement data. This provides a computable metric quantifying the "distance from freeness" as opposed to Saito's original binary criterion.

The practical computation of $\mathbb{P}^2$3 involves null space extraction, construction of a coefficient-space tensor $\mathbb{P}^2$4, and local minimization using alternating least squares (ALS) to optimize over the parameter pairs. The ALS approach leverages the bilinear structure of $\mathbb{P}^2$5 and is computationally efficient even for arrangements with up to $\mathbb{P}^2$6 lines.

Reinforcement Learning Framework for Free Arrangement Construction

Formulating arrangement synthesis as an MDP, the framework sequentially adds lines to an arrangement, with the agent's action space consisting of line selections from a large, finite candidate pool. The reward structure is hierarchical:

• Algebraic score: The primary reward is $\mathbb{P}^2$7, sharply peaked at free arrangements and smoothly interpolating the approach to freeness.
• Combinatorial pre-filtering: Fast arithmetic checks based on Betti numbers, intersection lattices, and Terao's exponent formula provide early gradient signals and filter out arrangements with impossible discriminants.
• Auxiliary signals: These include trajectory shaping for the Betti number, encouragement of high singularities (points of large multiplicity), penalties for pencil-type degeneracies, and an adaptive curriculum targeting diverse exponent types and arrangement sizes.

The agent's policy and value functions are parameterized by a Transformer-based actor-critic network, designed to exploit permutation symmetry and variable-length input sequences. Training employs PPO with curriculum sampling optimized for underrepresented sizes and exponent regimes, which is essential for avoiding trivial solutions and maximizing discovery of nontrivial free arrangements.

Numerical Results and Explicit Arrangements

The system efficiently discovers free arrangements, with exact symbolic verification when feasible. Explicit numerical results demonstrate the efficacy for $\mathbb{P}^2$8, producing arrangements with strongly nontrivial multiplicity profiles and exponents such as $\mathbb{P}^2$9 and $S$0. The results include line equations (often with rational coefficients) and detailed intersection multiplicity data, illustrating that the method identifies complex, highly structured arrangements that are otherwise impractical to enumerate or construct via traditional means.

Theoretical Implications and Future Directions

The functional $S$1 provides a new semicontinuous invariant with broader theoretical applications:

• Metric approach to freeness: It allows quantification of "proximity to freeness" and interpolates between combinatorially possible and algebraically realizable arrangements.
• Exploration of Terao's conjecture: By measuring $S$2 over the realization space of a fixed intersection lattice, one can empirically probe the invariance of freeness, offering computational tests for or against the conjecture.
• Extensions to higher dimensions and other types: The approach admits generalization to arrangements in higher-dimensional projective spaces, where Saito's criterion generalizes to multilinear forms. The methodology could also be adapted to investigate broader classes such as nearly-free or plus-one-generated arrangements by defining analogous relaxations for other algebraic criteria, e.g., Bourbaki degree.

While efficient, the current framework faces scaling challenges as the number and degree of candidate lines increases. Future improvements may leverage geometric priors, more advanced neural architectures, or more refined optimization techniques for the angular minimization task.

Conclusion

This work establishes a rigorous, geometric functional for semicontinuous relaxation of Saito's criterion and demonstrates its value both as a practical tool for the computer-aided synthesis of free line arrangements and as a new quantitative invariant in the study of arrangement freeness. The merger of algebraic geometry, numerical optimization, and machine learning offers a promising avenue for the systematic exploration of deep combinatorial and algebraic problems in arrangement theory and beyond.

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Reference:

"A semicontinuous relaxation of Saito's criterion and freeness as angular minimization" (2604.02995)