A penalised Saito functional for heuristic search of free line arrangements
Submitted:
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.