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OP-8761Status unassessedAnalysis

Log Concavity · Question 2

Stability, hyperbolicity, and zero localization of functions · Historical workshop question

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
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Problem statement

\label{fidk_conj} If p∈L-P+p\in\mathcal{L}\text{-}\mathcal{P}^+, then

∑k=0n∣akak+1ak+2ak−1akak+1ak−2ak−1ak∣xk∈L-P+.\sum_{k=0}^n \left|\begin{array}{ccc} a_k & a_{k+1} & a_{k+2} \\ a_{k-1} & a_k & a_{k+1}\\ a_{k-2} & a_{k-1} & a_k\end{array}\right|x^k\in\mathcal{L}\text{-}\mathcal{P}^+.

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Background

Workshop introduction

A list of the problems from the workshop on stability, hyperbolicity, and zero localization of functions, held December 5-9, 2011, is presented below. The problems have been loosely categorized for convenience, although many of the problems could be listed under several different headings.

Section context

Let L-P\mathcal{L}\text{-}\mathcal{P} denote the class of functions which are locally uniform limits of polynomials having only real zeros, and let the class of functions in L-P\mathcal{L}\text{-}\mathcal{P} which have only non-negative Maclaurin coefficients be denoted L-P+\mathcal{L}\text{-}\mathcal{P}^+.

Question context

The following conjecture of (McNamara et al.) was proved by P. Br"and'en. Let p(x):=∑k=0nakxkp(x):=\sum_{k=0}^n a_k x^k, with a−1:=an+1:=0a_{-1}:=a_{n+1}:=0. If p∈L-P+p\in\mathcal{L}\text{-}\mathcal{P}^+, then

∑k=0n∣akak+1ak−1ak∣xk∈L-P.\sum_{k=0}^n \left|\begin{array}{cc} a_k & a_{k+1} \\ a_{k-1} & a_{k} \end{array}\right|x^k \in\mathcal{L}\text{-}\mathcal{P}.

S. Fisk generalized this statement to polynomials with coefficients formed from the determinants of 3×33\times 3 matrices in the conjecture below.

Progress & source assessment

Historical workshop record. Current open/solved status has not been independently reassessed.

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References
  1. [1]Stability, hyperbolicity, and zero localization of functions link

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