A penalised Saito functional for heuristic search of free line arrangements

Abstract

We introduce the penalised Saito functional ${\mathfrak{S}}_{\lambda ,\beta }(\mathcal{A};d_1,d_2)$ for a reduced arrangement $\mathcal{A}$ of $n$ lines and a prescribed pair $d_1+d_2=n-1$. It measures the alignment of a candidate Saito determinant with the defining polynomial while penalising the failure of the candidate derivations to be logarithmic. We prove that the functional takes values in $[0,1]$, vanishes exactly when $\mathcal{A}$ is free with exponents $(1,d_1,d_2)$, and lies strictly between $0$ and $1$ otherwise. For fixed $(d_1,d_2)$, it is upper semicontinuous on the reduced configuration space, continuous at arrangements free with the prescribed pair, and converges as $\lambda \to \infty$ to the corresponding binary freeness test. We use a numerical approximation of this functional, together with a small $b_2$-shell term, to guide fixed-cardinality line-replacement searches over $\mathbb{Q}$ and selected quadratic extensions. Numerical values are used only to select candidates; every reported arrangement is certified in exact arithmetic using Saito's criterion. At the current snapshot, the certified database contains $6,146$ representatives with distinct Weisfeiler--Leman fingerprints and cardinalities up to $n=28$. Among them, $3,012$ have multiplicity gap $\epsilon (\mathcal{A})=d_1-m(\mathcal{A})\ge 2$, including lower-bound-extremal examples with $\epsilon =7$. These non-supersolvable arrangements provide test cases for studying realisation spaces and the persistence of freeness among realisations of the same intersection lattice, in connection with Terao's conjecture.

Substantially revised, with a new title. The former angular functional is replaced by a penalised Saito functional, with new proofs of its zero locus and semicontinuity. Section 3 and the search framework are rewritten. Results now include 6,146 exactly certified arrangements of up to 28 lines, including examples with multiplicity gap $\epsilon =7$