Matrix-analytic proof of the finite free Stam inequality

Develop a matrix-analytic proof of the finite free Stam inequality for the finite free convolution operator \boxplus_n acting on degree-n real-rooted polynomials, analogous to the derivative-case proof via Gauss–Lucas and differentiator matrices. Concretely, construct a matrix-analytic argument (using the Jacobian of the root map \Omega_{\boxplus_n} that sends root vectors (\alpha, \beta) to the roots of Poly(\alpha) \boxplus_n Poly(\beta)) that establishes the inequality (1/\Phi_n(p)) + (1/\Phi_n(q)) \le 1/\Phi_n(p \boxplus_n q) without relying on hyperbolic-polynomial convexity.

Background

In Section 4 the authors prove finite free Fisher information monotonicity under differentiation using a matrix-analytic approach grounded in explicit formulas for the Jacobian of the derivative root map and its relation to Gauss–Lucas weights and differentiator matrices. This yields sharp bounds via singular-value analysis.

In contrast, the proof of the finite free Stam inequality in Section 6 proceeds through a hyperbolicity-based approach using convexity results of Bauschke, Güler, Lewis, and Sendov, because analogous explicit Jacobian formulas for the finite free convolution root map \Omega_{\boxplus_n} are less tractable. The authors explicitly state that they were unable to apply these formulas to obtain a matrix-analytic proof and suggest it is reasonable to expect such a proof exists, potentially illuminating matrices in finite free position.

This open problem asks for an explicit matrix-analytic proof of the Stam inequality for \boxplus_n, paralleling the derivative-case method, by exploiting the Jacobian of the convolution root map and associated matrix structures.

References

When dealing with \boxplus_n, as in the case of \partial_x, it is still possible to obtain formulas for \Omega. However, these formulas are not as explicit as the one obtained in \Cref{lem:gauss_lucas_entries} and we did not see how to successfully apply them to get a proof. That said, it is reasonable to expect that there exists a matrix-analytic proof in the same spirit as the one provided in \Cref{sec:matrix_analysis_fisher_monotonicity}, which would potentially uncover interesting phenomena related to matrices in finite free position.

— Finite Free Information Inequalities  (2602.15822 - Garza-Vargas et al., 17 Feb 2026) in Section 6 (Stam's inequality via hyperbolicity), first paragraph

We conjecture that this defect controls the squared distance of the input velocity from the split locus, with a lower bound depending only on the degree and the relative input variances, even as individual root gaps tend to zero.

— When Finite Free Curves Split  (2609.10367 - Hashemi et al., 9 Sep 2026) in Conjecture 1, Section 1.4 ('Further questions')

Can \Cref{thm:tangent-convolution} be extended to successive rescalings of their colliding clusters so as to identify the leading convolution at each separation scale? Can these descriptions establish eq:quantitative-split-rigidity along such degenerations, with a constant depending only on n?

eq:quantitative-split-rigidity:

∥w∥22−∥Jw∥22≥cn 4VpVq(Vp+Vq)2dist⁡(w,Lα,β)2.\|w\|_2^2-\|Jw\|_2^2 \geq c_n\,\frac{4V_pV_q}{(V_p+V_q)^2} \operatorname{dist}(w,L_{\alpha,\beta})^2.

— When Finite Free Curves Split  (2609.10367 - Hashemi et al., 9 Sep 2026) in Question 1, Section 1.4 ('Further questions')