Is there an example of a Cohen-Macaulay local domain $R$ of characteristic $p>0$ for which the Hilbert-Kunz multiplicity $e\_{HK}(R)$ is *not* equal to its Hilbert-Samuel multiplicity $e(R)$? If no example, a result that states they may not in general be equal would also be helpful, of course. 



Hilbert-Kunz multiplicity of Cohen-Macaulay local domains
In general $e\_{HK}(R) \leq e(R)$. Most of the time the inequality is strict.


The case $e(R)=2$ and $\dim R=2$ is studied carefully in the paper by Yoshida-Watanabe "Hilbert-Kunz multiplicity of two-dimensional local rings" , [available here](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-162/issue-none/Hilbert-Kunz-multiplicity-of-two-dimensional-localrings/nmj/1114631592.full).


For example, in this case $e\_{HK}<2$ if $R$ is a complete rational double point. In this case one can write $R = k[[x,y,z]]^G$ (assume $k$ alg. closed with big enough characteristic), and it turns out that $e\_{HK}(R) = 2-1/|G|<2=e(R)$ (see Example 4.1).


Of course, there are many more results on both kinds of multiplicities, but $e\_{HK}$ is *much* more difficult to compute. Paul Monsky is really good at computing them (among many other things), so you can check out his papers for many exotic examples.


**Added**: Of course, this being MO, Paul himself has added via the comment below some very interesting and relevant results. One thing that may help newcomers: in his comment we look at homogenous hypersurfaces of degree $d$. Those rings automatically have $e(R)=d$. So for example the work by Brenner-Trivedi (not joint) says that $e\_{HK}<e$ when there are $3$ variables, $d=4$ and the polynomial is absolutely irreducible.


