Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraicity of the Brascamp-Lieb constants

Calin Chindris, Harm Derksen

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.

Algebraicity of the 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.
Paper Structure (4 sections, 11 theorems, 71 equations)

This paper contains 4 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The capacity of geometric data
  3. Proof of Theorem
  4. The capacity as a function on the entire set of feasible data

Key Result

Theorem 1

Let Then $\mathbf{cap}_\mathcal{Q}(-, \mathbf p): \bm{\mathscr{V}} \to \mathbb R$ is a semi-algebraic function. Consequently, it is an algebraic function in the sense that there exists a non-zero polynomial $P$ in $1+\sum_{i,j}|\mathscr{A}_{ij}|d_i n_j$ variables such that

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • Remark 6
  • proof : Proof of Theorem
  • Corollary 7
  • ...and 12 more