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We prove the existence of a compactly supported, continuous (except at finitely many points) function gI,m:[0,∞)⟶R for all monomial prime ideals I of R of height one where (R,m) is the homogeneous coordinate ring associated to a projectively normal toric pair (X,D), such that ∫0∞gI,m(λ)dλ=β(I,m), where β(I,m) is the second coefficient of the Hilbert-Kunz function of I with respect to the maximal ideal m, as proved by Huneke-McDermott-Monsky [8]. Using the above result, for standard graded normal affine monoid rings we give a complete description of the class map τm:Cl(R)⟶R introduced in [8] to prove the existence of the second coefficient of the Hilbert-Kunz function. Moreover, we show the function gI,m is multiplicative on Segre products with the expression involving the first two coefficients of the Hilbert plolynomial of the rings and the ideals. © 2021 Elsevier B.V.

Department of Mathematics

Fulltext

Journal of Pure and Applied Algebra

β-Density function on the class group of projective toric varieties

Let p be a prime number, k a field of characteristic p and G a finite p-group. Let V be a finite-dimensional linear representation of G over k. Write S = Sym V ∗. For a class of p-groups which we call generalised Nakajima groups, we prove the following: (a) The Hilbert ideal is a complete intersection. As a consequence, for the case of generalised Nakajima groups, we prove a conjecture of Shank and Wehlau (reformulated by Broer) that asserts that if the invariant subring SG is a direct summand of S as SG-modules then SG is a polynomial ring. (b) The Hilbert ideal has a generating set with elements of degree at most |G|. This bound is conjectured by Derksen and Kemper. 2021 American Mathematical Society

World University root node

Proceedings of the American Mathematical Society

ON HILBERT IDEALS for A CLASS of p-GROUPS in CHARACTERISTIC p

We prove that, analogous to the Hilbert–Kunz density function, (used for studying the Hilbert–Kunz multiplicity, the leading coefficient of the Hibert–Kunz function), there exists a β-density function gR , m: [0 , ∞) ⟶ R, where (R, m) is the homogeneous coordinate ring associated with the toric pair (X, D), such that ∫0∞gR,m(x)dx=β(R,m),where β(R, m) is the second coefficient of the Hilbert–Kunz function for (R, m) , as constructed by Huneke–McDermott–Monsky. Moreover, we prove, (1) the function gR , m: [0 , ∞) ⟶ R is compactly supported and is continuous except at finitely many points, (2) the function gR , m is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk–Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Journal of Algebraic Combinatorics

Density function for the second coefficient of the Hilbert–Kunz function on projective toric varieties

For a toric pair (X,D), where X is a projective toric variety of dimension d−1≥1 and D is a very ample T-Cartier divisor, we show that the Hilbert–Kunz density function HKd(X,D)(λ) is the d−1 dimensional volume of P‾D∩{z=λ}, where P‾D⊂Rd is a compact d-dimensional set (which is a finite union of convex polytopes). We also show that, for k≥1, the function HKd(X,kD) can be replaced by another compactly supported continuous function φkD which is ‘linear in k’. This gives the formula for the associated coordinate ring (R,m): limk→∞⁡[Formula presented]=[Formula presented]∫0∞φD(λ)dλ where φD (see Proposition 1.2) is solely determined by the shape of the polytope PD, associated to the toric pair (X,D). Moreover φD is a multiplicative function for Segre products. This yields explicit computation of φD (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope PD, one can explicitly compute the limit for two dimensional toric pairs and their Segre products. We further show that (Theorem 6.3) the renormalized limit takes the minimum value if and only if the polytope PD tiles the space MR=Rd−1 (with the lattice M=Zd−1). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope. © 2018 Elsevier Inc.