Preservation of inequalities under Hadamard products

TL;DR

This work investigates how Hadamard products affect key combinatorial inequalities captured by the numerator polynomials $\mathscr{W}(p)$ of generating functions. Using the Lorentzian-polynomial framework, it proves that ultra log-concavity is preserved under Hadamard products and that $\gamma$-positivity is maintained for symmetric factors with equal defect, with further preservation results for various symmetric decompositions, including interlacing. It also provides a striking counterexample to a conjecture on Hadamard powers, showing that even with nonnegative $\mathscr{W}(p)$, the polynomials $\mathscr{W}(p^n)$ need not be real-rooted (or log-concave) for any $n\ge1$, via Ehrhart-theoretic constructions such as the Reeve tetrahedron. Collectively, the paper delineates both the robust and fragile aspects of Hadamard-product preservation, connecting Lorentzian polynomial theory with Ehrhart theory and offering avenues for further refinement of when these combinatorial properties endure under Hadamard operations.

Abstract

Wagner (1992) proved that the Hadamard product of two Pólya frequency sequences that are interpolated by polynomials is again a Pólya frequency sequence. We study whether related combinatorial properties are preserved under Hadamard products. In particular, we show that ultra log-concavity, $γ$-positivity, and interlacing symmetric decompositions are preserved. Furthermore, we disprove a conjecture by Fischer and Kubitzke (2014) concerning the real-rootedness of Hadamard powers.

Theorems & Definitions (51)

• Theorem 1.1: wagner
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5: fischer-kubitzke
• Theorem 1.6
• Theorem 2.1: branden-huh
• Theorem 2.2
• Lemma 2.3