Dimension Bounds for Systems of Equations with Graph Structure

• DOI: 10.48550/arxiv.2410.06840
• Publication Date: 2024-10-09
• Authors: Eddie Nijholt; Davide Sclosa
• Keywords: FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); Dynamical Systems (math.DS); Mathematics - Dynamical Systems; 05E99 (Primary), 05C50, 34A34 (Secondary)

We introduce a broad class of equations that are described by graph, which includes many well-studied systems. For these, we show the number solutions (or dimension solution set) can be bounded studying certain induced subgraphs. As corollaries, obtain novel bounds in spectral graph theory on multiplicities eigenvalues, and nonlinear dynamical system equilibrium set network.