### Tight Hilbert Polynomials

Keywords

Abstract In this talk, we will explore the Hilbert function and the Hilbert polynomial of filtrations of ideals arising from the tight closure of ideals. We compute the tight Hilbert polynomial in some diagonal hypersurface rings. In most cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. Let $(R, m)$ be a Noetherian local ring of prime characteristic $p$, and $Q$ be an $m$-primary parameter ideal. We provide criteria for $F$-rationality of $R$ using the tight Hilbert function $H_Q^\ast(n) = \ell(R/(Q^n)^\ast)$ and the coefficient $e_1^\ast(Q)$ of the tight Hilbert polynomial $P_Q^\ast(n) = \sum_{i=0}^{d} (-1)^i e_i^\ast(Q)$. Craig Huneke asked if the $F$-rationality of unmixed local rings may be characterized by the vanishing of $e_1^\ast(Q)$. We construct examples to show that without additional conditions, this is not possible. Let $R$ be an excellent, reduced, equidimensional Noetherian local ring, and $Q$ be generated by parameter test elements. We find formulas for $e_1^\ast(Q), e_2^\ast(Q), \ldots, e_d^\ast(Q)$ in terms of Hilbert coefficients of $Q$, lengths of local cohomology modules of $R$, and the length of the tight closure of the zero submodule of $H^d_m(R)$. Using these, we prove:

$R$ is $F$-rational iff $e_1^\ast(Q) = e_1(Q)$ iff $\text{depth} R >= 2$ and $e_1^\ast(Q) = 0$.

Let $I$ be an ideal generated by a system of parameters in an excellent Cohen-Macaulay local domain then the associated graded ring $G^\ast(I)$ of the filtration $\{ (I^n)^\ast \}$ is Cohen-Macaulay.

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