We define a function, called s-multiplicity, that interpolates between Hilbert–Samuel multiplicity and Hilbert–Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert–Samuel multiplicity for small values of s and is equal to Hilbert–Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert–Samuel and Hilbert–Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real space.
W.D. Taylor "Interpolating between Hilbert–Samuel and Hilbert–Kunz multiplicity" Journal of Algebra, Volume 509, 2018, Pages 212-239, ISSN 0021-8693, https://doi.org/10.1016/j.jalgebra.2018.05.015.