**Vijaylaxmi Trivedi**

TIFR

March 10, 2016

**Hilbert-Kunz density function and Hilbert-Kunz multiplicity**:
In this talk we recall a well-studied ${
m char}~p$ invariant
{it Hilbert-Kunz multiplicity}, $e_{HK}(R, I)$, for a local ring/standard graded ring $R$ with respect to an {f m}-primary/graded ideal of finite colength $I$. This could be considered as an analogue of Hilbert-Samuel function and Hilbert-Samuel multiplicity (but specific to characteristic $p > 0$).
We give a brief survey of some of the results on this invariant and
try to convey why $e_{HK}$ is a `better' and a `worse' invariant than
Hilbert-Samuel multiplicity of a ring.
In the graded case (based on the recent work), for a pair $(R, I)$
we introduce a new invariant, the {it Hilbert-Kunz density function},
which is a limit of a uniformly convergent sequence of real valued compactly supported, piecewise linear and continous functions. We express $e_{HK}(R, I)$ as an integral of this function.
We prove that this function (unlike $e_{HK}$) satisfies a multiplication formula for the Segre product of rings. As a conseque