Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ideal of $\mathcal{O}_K$. I want to get a upper bound of the local Hilbert-Samuel function of $X$ at the point $\xi$, the upper bound is related to the degree of $X$. Attention: $\xi$ may not be regular.
I know such a result for $X$ is a general projective variety which published is JAG of 1997, but it is too coarse, and I want a better approximation.
By the way, if I know a approximation better enough, can I use the Koszul-complexes to estimate the case of complete intersection?