Density functions for epsilon multiplicity and families of ideals

Abstract A density function for an algebraic invariant is a measurable function on which measures the invariant on an ‐scale. This function carries a lot more information related to the invariant without seeking extra data. It has turned out to be a useful tool, which was introduced by the third author in Trivedi [Trans. Amer. Math. Soc. 370 (2018), no. 12, 8403–8428], to study the characteristic invariant, namely Hilbert–Kunz multiplicity of a homogeneous ‐primary ideal. Here, we construct density functions for a Noetherian filtration of homogeneous ideals and for a filtration given by the saturated powers of a homogeneous ideal in a standard graded domain . As a consequence, we get a density function for the epsilon multiplicity of a homogeneous ideal in . We further show that the function is continuous everywhere except possibly at one point, and is a continuous function everywhere and is continuously differentiable except possibly at one point. As a corollary, the epsilon density function is a compactly supported continuous function on except at one point, such that . All the three functions , and remain invariant under passage to the integral closure of . As a corollary of this theory, we observe that the ‘rescaled’ Hilbert–Samuel multiplicities of the diagonal subalgebras form a continuous family.