Inequalities for rank-two permanents and finite free convolutions
Abstract
Bang (1976) proved the inequality for matrix permanents per2(A) ≥ 2−2nper(A ⊗ J2), where J2 is the 2 × 2 all-ones matrix and A is any n × n matrix with non-negative entries. We show that, if A is any n × n real-valued matrix with rank at most two (possibly having negative entries), this inequality can be sharpened, replacing the constant 2−2n by 1 / binom2nn = (n!)2 / (2n)! > 2−2n. We then show that this sharpened inequality also implies new inequalities for finite free convolutions of polynomials: if p and q are monic real-rooted polynomials of degree n, then (p boxplusn q)(x)2 ≥ (p2 boxplus2n q2)(x) and (p boxtimesn q)(x)2 ≥ (p2 boxtimes2n q2)(x) for all x ∈ ℝ, for boxplusn and boxtimesn the finite free additive and multiplicative convolution operations, respectively, on polynomials of degree n.
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BibTeX
@misc{kunisky2026inequalities,
title = {{Inequalities for rank-two permanents and finite free convolutions}},
author = {Dmitriy Kunisky and Daniel A. Spielman and Xifan Yu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28520},
eprint = {2608.28520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28520},
}
RIS
TY - UNPB
TI - Inequalities for rank-two permanents and finite free convolutions
AU - Dmitriy Kunisky
AU - Daniel A. Spielman
AU - Xifan Yu
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28520
UR - https://arxiv.org/abs/2608.28520
AB - Bang (1976) proved the inequality for matrix permanents per^2(A) ≥ 2^−2nper(A ⊗ J_2), where J_2 is the 2 × 2 all-ones matrix and A is any n × n matrix with non-negative entries. We show that, if A is any n × n real-valued matrix with rank at most two (possibly having negative entries), this inequality can be sharpened, replacing the constant 2^−2n by 1 / binom2nn = (n!)^2 / (2n)! > 2^−2n. We then show that this sharpened inequality also implies new inequalities for finite free convolutions of polynomials: if p and q are monic real-rooted polynomials of degree n, then (p boxplus_n q)(x)^2 ≥ (p^2 boxplus_2n q^2)(x) and (p boxtimes_n q)(x)^2 ≥ (p^2 boxtimes_2n q^2)(x) for all x ∈ ℝ, for boxplus_n and boxtimes_n the finite free additive and multiplicative convolution operations, respectively, on polynomials of degree n.
ER -