For a toric pair $(X, D)$, where $X$ is projective variety of dimension $d-1\geq 1$ and $D$ very ample $T$-Cartier divisor, we show that the Hilbert-Kunz density function $HKd(X, D)(\lambda)$ $d-1$ dimensional volume ${\overline {\mathcal P}}_D \cap \{z= \lambda\}$, P}}_D\subset {\mathbb R}^d$ compact $d$-dimensional set (which finite union convex polytopes). We also that, for $k\geq 1$, kD)$ can be replaced by another compactly supported continuous $\varphi_{kD}$ which `linear in $k$'. This...

We prove that, analogous to the HK density function, (used for studying Hilbert-Kunz multiplicity, leading coefficient of function), there exists a $\beta$-density function $g_{R, {\bf m}}:[0,\infty)\longrightarrow {\mathbb R}$, where $(R, m})$ is homogeneous coordinate ring associated toric pair $(X, D)$, such that $$\int_0^{\infty}g_{R, m}}(x)dx = \beta(R, m}),$$ $\beta(R, second m})$, as constructed by Huneke-McDermott-Monsky. Moreover we prove, (1) m}}:[0, \infty)\longrightarrow R}$...

Let $\Bbbk$ be a field of characteristic $p>0$, $V$ finite-dimensional $\Bbbk$-vector-space, and $G$ finite $p$-group acting $\Bbbk$-linearly on $V$. $S = \Sym V^*$. We show that $S^G$ is polynomial ring if only the dimension its singular locus less than $\rank_\Bbbk V^G$. Confirming conjecture Shank-Wehlau-Broer, we direct summand $S$, then ring, in following cases: \begin{enumerate} \item $\Bbbk \bbF_p$ V^G 4$; or $|G| p^3$. \end{enumerate} In order to prove above result, also \geq...

For a standard graded ring $R$ of dimension $\geq 2$ over perfect field characteristic $p>0$ and homogeneous ideal $I$ finite colength, the HK density function with respect to is compactly supported continuous $f_{R, I}:[0, \infty)\longto [0, \infty)$, whose integration yields \mbox{HK} multiplicity $e_{HK}(R, I)$. Here we answer question V. Trivedi about Hilbert-Kunz tensor product rings show that it convolution factor rings. Using Fourier transform, as corollary get We compute transform...

Let $p$ be a prime number, $\Bbbk$ field of characteristic and $G$ finite $p$-group. $V$ finite-dimensional linear representation over $\Bbbk$. Write $S = \mathrm{Sym} V^*$. For class $p$-groups which we call generalised Nakajima groups, prove the following: \begin{enumerate} \item The Hilbert ideal is complete intersection. As consequence, for case conjecture Shank Wehlau (reformulated by Broer) that asserts if invariant subring $S^G$ direct summand $S$ as $S^G$-modules then polynomial...