A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite

Abstract: We prove that every finite multivariate Gaussian mixture density has only finitely many modes. Our approach combines an algebraic formulation of the ridgeline theory of Ray and Lindsay (2005) with a transcendence-degree argument based on Ax's functional-transcendence theorem to bound the cardinality of the set of critical points. Our techniques extend recent work by Wang (2026), who used Ax's theorem together with real-analytic curve selection to prove finiteness of the critical set of homoscedastic Gaussian mixtures. We introduce the ridgeline correspondence and use it to obtain a finiteness result that applies to arbitrary heteroscedastic Gaussian mixtures. Our framework also establishes finiteness of the number of modes for additional classes of polynomial-exponential mixtures and generalized Gaussian mixtures.