An algebraic theory of Lojasiewicz exponents

Abstract

We develop a unified algebraic and valuative theory of Lojasiewicz exponents for pairs of graded families and filtrations of ideals. Within this framework, local Lojasiewicz exponents, gradient exponents, and exponents at infinity are all realized as asymptotic containment thresholds between filtrations, governed by integral closure. This reformulation shows that Lojasiewicz exponents are fundamentally valuative optimization problems. The central structural contribution of the paper is a finite-max principle. Under verifiable algebraic hypotheses, the a priori infinite valuative supremum bounding the Lojasiewicz exponent reduces to a finite maximum, and computes the Lojasiewicz exponent precisely. We identify two complementary mechanisms leading to this phenomenon: finite testing arising from normalized blowups and Noetherian Rees algebras, and attainment via compactness of normalized valuation spaces under linear boundedness assumptions. This finite-max framework yields strong structural consequences. We prove rigidity results showing that common extremal valuations force equality of Lojasiewicz ratios, and we establish stratification and stability phenomena for Lojasiewicz exponents in families, including fractional linearity and wall-chamber behavior along natural one-parameter deformations. The theory recovers and explains classical results in toric and Newton-polyhedral settings, particularly, for Newton nondegenete case, where the Lojasiewicz exponent is computed by finitely many toric divisorial valuations corresponding to facet data. Finally, we illustrate why the hypotheses underlying the finite-max principle are essential, delineating the precise scope of the theory.