Estimates for the characteristic function of a prime ideal

Let $k$ be a field of characteristic 0, $\mathfrak p$ a homogeneous prime ideal of the ring $k[X]=k[x_0,\dots,x_m]$ $(m\geqslant1)$ and $\mathfrak L_\mathfrak p(\nu)$ the set of residues of homogeneous polynomials of degree $\nu$ ($\nu$ is a natural number) in $k[X]$, taken modulo $\mathfrak p$. In this paper an inequality is proved for the dimension of the vector space $\mathfrak L_\mathfrak p(\nu)$ which is valid for $\nu\geqslant1$.
Bibliography: 6 titles.