A penalised Saito functional for heuristic search of free line arrangements

arXiv:2604.02995v3 Announce Type: replace-cross 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 $\epsilon(\mathcal{A})=d_1-m(\mathcal{A})\geq2$, 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.

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 $\epsilon(\mathcal{A})=d_1-m(\mathcal{A})\geq2$, 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 $\epsilon=7$

Algebraic Geometry (math.AG); Machine Learning (cs.LG); Combinatorics (math.CO)

arXiv:2604.02995 [math.AG]

(or arXiv:2604.02995v3 [math.AG] for this version)

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A penalised Saito functional for heuristic search of free line arrangements

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