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

arXiv:2604.02995v2 Announce Type: replace-cross Abstract: We introduce a nonnegative functional mathfrak{S} on the space of line arrangements in P^2 that vanishes precisely on free arrangements, obtained as a semicontinuous relaxation of Saito's criterion. Given an arrangement 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 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(A) in coefficient space, providing a computable measure of how far an arrangement is from admitting a free basis of logarithmic derivations of the expected degrees. We prove that mathfrak{S} is upper semicontinuous on natural strata, and use this to give a functional reformulation of Terao's conjecture. Beyond its theoretical interest, mathfrak{S} provides a viable computational handle on the landscape of free arrangements. We illustrate this through two complementary roles: as a smooth reward signal driving a reinforcement learning search for moderate n, and as a fast pre-filter accelerating an algebraic extension procedure for larger n. For n leq 13, the reinforcement learning system discovers hundreds of verified free arrangements spanning all admissible exponent types. For n geq 14, where the reinforcement learning reward signal becomes insufficient, the hybrid extension procedure -- combined with classical supersolvable constructions -- produces at least one verified free arrangement for every admissible exponent pair (d_1, d_2) with n leq 20.

Source: arXiv cs.LG | 2026-05-07