Let (R, m) be an analytically unramified local ring of positive prime characteristic p.For an ideal I, let I * denote its tight closure. We introduce the tight Hilbert function H * I (n) = ℓ(R/(I n ) * ) and the corresponding tight Hilbert polynomial P * I (n), where I is an m-primary ideal. It is proved that F -rationality can be detected by the vanishing of the first coefficient of P * I (n). We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes. IntroductionLet (R, m) be a d-dimensional Noetherian local ring and I be an m-primary ideal. Let I be the integral closure of I. The Rees algebra of I is denoted byWe use ℓ(M ) to denote the length of an R-module M. David Rees [14] showed that if R is analytically unramified then R(I) is a finite module over R(I). This implies that for all large n, the normal Hilbert function of I, H I (n) = ℓ(R/I n ) is a polynomial of degree d. This is called the normal Hilbert polynomial of I and it is denoted by P I (n). We write P I (n) = e 0 (I)Here e 0 (I) = e(I), the multiplicity of I and the coefficients e i (I) for i = 0, 1, . . . , d are called the normal Hilbert coefficients of I. The normal Hilbert coefficients play an important role in the study of singularities of algebraic varieties. Rees [15] proved that if (R, m) is a 2-dimensional analytically unramified normal local ring then it is pseudo-rational if and only if e 2 (I) = 0 for all m-primary ideals I of R. Shiroh Itoh [9] proved that if R is Cohen-Macaulay and analytically unramified, then e 3 (I) ≥ 0. Moreover, if R is Gorenstein and I is generated by a regular sequence so that I = m, then e 3 (I) = 0 if and only if r(I) ≤ 2. Here r(I) = min{n | II n = I n+1 }. The integer r(I) is called the normal reduction number of I. Itoh [9] proved that if r(I) ≤ 2, then R(I) is Cohen-Macaulay. He

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In this expository paper, we present simple proofs of the Classical, Real, Projective and Combinatorial Nullstellensätze. Several applications are also presented such as a classical theorem of Stickelberger for solutions of polynomial equations in terms of eigenvalues of commuting operators, construction of a principal ideal domain which is not Euclidean, Hilbert's 17 th problem, the Borsuk-Ulam theorem in topology and solutions of the conjectures of Dyson, Erdös and Heilbronn. * This article grew out of discussions with late Prof. Dr. Uwe Storch (1940Storch ( -2017 and lectures delivered by the second and the third author in various workshops and conferences. Prof. Uwe Storch was known for his work in commutative algebra, analytic and algebraic geometry, in particular derivations, divisor class group and resultants.

Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}({\mathfrak {n}}) be the Rees algebra of the maximal homogeneous ideal n {\mathfrak {n}} of R . R. We show that the generalized Hilbert-Kunz function H K ( s ) = ℓ ( R ( n ) / ( n , n t ) [ s ] ) HK(s)=\ell ({\mathcal {R}}({\mathfrak {n}})/({\mathfrak {n}}, {\mathfrak {n}} t)^{[s]}) is given by a polynomial for all large s . s. We calculate it in many examples and also provide a Macaulay2 code for computing H K ( s ) . HK(s).

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