Related papers: Some computations of generalized Hilbert-Kunz func…
The Hilbert-Kunz multiplicity and $F$-signature are important invariants for researchers in commutative algebra and algebraic geometry. We provide software, and describe the automation of a calculation, for the two invariants in the case of…
For a standard graded ring $R$ of dimension $\geq 2$ over a perfect field of characteristic $p>0$ and a homogeneous ideal $I$ of finite colength, the HK density function of $R$ with respect to $I$ is a compactly supported continuous…
We prove that, under certain assumptions, generalized Hilbert-Kunz multiplicities can be expressed as linear combinations of classical Hilbert-Kunz multiplicities.
We consider a finite dimensional $\kk G$-module $V$ of a $p$-group $G$ over a field $\kk$ of characteristic $p$. We describe a generating set for the corresponding Hilbert Ideal. In case $G$ is cyclic this yields that the algebra $\kk[V]_G$…
We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic.…
This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and…
Let F be a finite field of characteristic 2 and h be the element x^3+y^3+xyz of F[[x,y,z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x^q,y^q,z^q,h^j) for q a power of 2. We also…
We determine the Hilbert-Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this…
Given a commutative local ring $(R,\mathfrak m)$ and an ideal $I$ of $R$, a family of quotients of the Rees algebra $R[It]$ has been recently studied as a unified approach to the Nagata's idealization and the amalgamated duplication and as…
Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and ${\mathcal R}(\mathfrak{n})$ be the Rees algebra of the maximal homogeneous ideal $\mathfrak{n}$ of $R.$ We show that the generalized Hilbert-Kunz function…
In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher $\tor$ counterparts $\i t_i=\underset{n \to \infty}{\lim} \l(\tor_i(k,{}^{f^n}…
We discuss Hilbert-Kunz function from when it was originally defined to its recent developments. A brief history of Hilbert-Kunz theory is first recounted. Then we review several techniques involved in the study of Hilbert-Kunz functions by…
A result of Monsky states that the Hilbert-Kunz function of a one-dimensional local ring of prime characteristic has a term $\phi$ that is eventually periodic. For example, in the case of a power series ring in one variable over a…
In this article, I provide a solution to a rank computation problem related to the computation of the Hilbert-Kunz function for any disjoint-term trinomial hypersurface, over any field of characteristic 2. This rank computation problem was…
The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the…
The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings.
Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in $m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2 \times 2$ minors of the generic $m \times…
The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber product rings and Nagata idealizations. We give explicit formulas for the Hilbert-Kunz multiplicity of a fiber product $R \times_T S$, where…
Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2…
We develop Hilbert-Kunz theory in a combinatorial setting namely for binoids. We show that the Hilbert-Kunz multiplicity for commutative, finitely generated, semipositive, cancellative and reduced binoids exists and is a rational number.…