Algebraicity of the Brascamp-Lieb constants

TL;DR

This paper proves that the Brascamp-Lieb constant is a semi-algebraic and algebraic function of the data, extending the result to quiver Brascamp-Lieb constants, revealing their polynomial relations.

Contribution

It establishes the algebraic nature of the Brascamp-Lieb constants and generalizes the result to quiver representations, providing new insights into their structure.

Findings

BL constant is semi-algebraic on feasible data

BL constant satisfies a polynomial relation

Extension to quiver Brascamp-Lieb constants

Abstract

We show that the Brascamp-Lieb (BL) constant BL(-,p) is a semi-algebraic function on the set of feasible data. Consequently, it is algebraic in the sense that it satisfies a polynomial relation of the form P(V, BL(V,p))=0 for a non-zero polynomial P. In fact, we establish an analogous statement in the more general setting of quiver BL constants associated to representations of bipartite quivers.