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Graphons are symmetric measurable functions that arise from a sequence ofgraphs. A graphon variety is the a set of all graphons defined by a conditionof the form $t(g, W) = 0$ for a fixed quantum graph $g$, where $t(.,.)$ is thehomomorphism density and a quantum graph is a formal linear combination ofmultigraphs. Using a method of representing graphs as polynomials, we constructan epimorphism from the space of quantum graphs to a subring of the complexpolynomial ring that is invariant under permutations of variables. Whengraphons 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 wecan call such sets closed under the Zariski Topology. We determine several tiesto Algebraic Geometry as a result of utilizing finite rank kernels and discoverthat a weaker version of Hilbert's Nullstellensatz applies to kernel zero-setswith respect to homomorphism density. Throughout, we examine the connectionbetween Algebraic Geometry and Graphon Theory.

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