A Dittert-like inequality for homogeneous stable polynomials
2026-10-29
Research publication
A Dittert-like inequality for homogeneous stable polynomials
Preprint
- Original date
- Authors
- Yair Lavi
- Combinatorics
- Matrix theory
- Permanents
- Real stable polynomials
Abstract
We prove a coefficient inequality for every homogeneous real stable polynomial \(P(s,x_1,\ldots,x_n)\) of degree \(n\ge1\) with nonnegative coefficients. If \(P(1,0,\ldots,0)=1\), \(r_i=\partial_{x_i}P(1,0,\ldots,0)\), and \(\sum_i r_i=n\), the following inequality holds, where brackets denote coefficient extraction:
\[ [x_1\cdots x_n]P(0,x_1,\ldots,x_n)+2-\prod_i r_i-P(0,1,\ldots,1)\ge\frac{n!}{n^n}. \]
Equality holds exactly for \(P=(s+\sum_i x_i/n)^n\). The normalizations can be removed by rescaling \(P\) and the \(x_i\). We give applications to permanents, mixed discriminants, subpermanents, and stable combinatorial measures. In particular, the theorem gives an independent proof of Dittert’s conjecture for nonnegative matrices in every dimension.