Finite rank kernel varieties: A variant of Hilbert's Nullstellensatz for graphons and applications to Hadamard matrices

TL;DR

This paper explores the algebraic structure of graphons, especially finite rank ones, using polynomial representations and algebraic geometry concepts, leading to a Nullstellensatz analogue for kernel zero-sets.

Contribution

It introduces a polynomial framework for graphon varieties, establishes an algebraic kernel set concept, and applies a Nullstellensatz analogue to finite rank graphons.

Findings

Finite rank graphons have algebraic kernel sets that are Zariski-closed.

A Nullstellensatz analogue applies to kernel zero-sets in this context.

The work links algebraic geometry principles to graphon theory.

Abstract

Graphons are symmetric measurable functions that arise from a sequence of graphs. A graphon variety is the a set of all graphons defined by a condition of the form $t(g, W) = 0$ for a fixed quantum graph $g$, where $t(.,.)$ is the homomorphism density and a quantum graph is a formal linear combination of multigraphs. Using a method of representing graphs as polynomials, we construct an epimorphism from the space of quantum graphs to a subring of the complex polynomial ring that is invariant under permutations of variables. When graphons are of finite rank, we demonstrate that an analog of the "ideal" inverse in Algebraic Geometry is an ideal in our polynomial representation. Defining an algebraic kernel set using kernel varieties, we demonstrate that we can call such sets closed under the Zariski Topology. We determine several ties to Algebraic Geometry as a result of utilizing finite rank kernels and discover that a weaker version of Hilbert's Nullstellensatz applies to kernel zero-sets with respect to homomorphism density. Throughout, we examine the connection between Algebraic Geometry and Graphon Theory.