Density functions for some algebraic invariants

A density function for an algebraic invariant is a measurable function on R which measures the invariant on an R-scale, that is, a density function for a given algebraic invariant, say , is an integrable function f: R −→ R, which gives a new measure μf on R such that the integration R E f on a subset E ⊂ R is ‘the measure of the invariant  on E’.
This function carries a lot more information related to the invariant without seeking extra data (than needed to study the invariant itself). The HK density function, which was introduced by the speaker to study Hilbert-Kunz multiplicity, turned out to be a useful tool in answering some long standing open questions. In this talk we discuss the existence of density function for some well known and some elusive algebraic invariants and its applications. A part of the talk is based on some work with K.I. Watanabe, and some with S.Das and S.Roy.

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