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arXiv · 2609.33651

Log-concavity of flat arrangement polynomials

Yuan Gao·Tianyu Yuan
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Abstract

We prove log-concavity for the determinant-weighted external semi-activity polynomials of all real flat arrangements, strengthening their known trapezoidality. In fact, we establish a quadratic coefficient inequality that, in rank at least two, implies power concavity with an explicit rank-dependent exponent. The proof uses a new mixed-volume representation of the coefficients and the Alexandrov--Fenchel inequality. A more general formula gives a factorization and log-concavity for related mixed-volume sequences. As applications, we establish the conjectured log-concavity for spanning-tree polynomials of Eulerian digraphs, extend it to positive circulation weights, and strengthen the coefficient inequalities for Alexander polynomials of special alternating links.

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Yuan Gao, Tianyu Yuan. 2026-09-27. Log-concavity of flat arrangement polynomials. https://arxiv.org/abs/2609.33651

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