Nonlinear Algebra for Interaction Networks

Many scientific systems are driven by interactions among their components. We study interaction dynamics through the generalized Lotka--Volterra (GLV) equation, where interaction rates define a directed network through their sign pattern. Feasibility and stability become nonlinear-algebraic conditions: they define semialgebraic sets and can be studied via Grassmannians, leading to computational tests for “impossible” networks through sign-realization. We also discuss two directions: tensorial GLV models for higher-order (hypergraph) interactions, and antisymmetric cases such as the Volterra lattice (the three-species rock–paper–scissors model), where solutions are described by algebraic curves and Riemann theta functions, with tropical degenerations linking them to graph combinatorics. The first part is based on arXiv:2509.00165; the second reports ongoing work with G. Almeida building on arXiv:2512.13366.