[Paper Review] Hilbert-Kunz functions of 2 x 2 determinantal rings

Lance Edward Miller, Irena Swanson|arXiv (Cornell University)|Jun 5, 2012

Graph theory and applications12 references3 citations

TL;DR

This paper develops a recursive Gröbner basis-based method to compute the generalized Hilbert-Kunz function of 2×2 determinantal rings over any field. It proves the function is polynomial in q and provides closed-form expressions for m ≤ 2, yielding characteristic-independent Hilbert-Kunz multiplicities and confirming polynomial behavior for these rings.

ABSTRACT

Let k be an arbitrary field (of arbitrary characteristic) and let X = [x_{i,j}] be a generic m x n matrix of variables. Denote by I_2(X) the ideal in k[X] = k[x_{i,j}: i = 1, ..., m; j = 1, ..., n] generated by the 2 x 2 minors of X. We give a recursive formulation for the lengths of the k[X]-module k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q)) as q varies over all positive integers using Grobner basis. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We give closed forms for the cases when m is at most 2, %as well as the closed forms for some other special length functions. We apply our method to give closed forms for these Hilbert-Kunz functions for cases $m \le 2$.

Motivation & Objective

• To compute the generalized Hilbert-Kunz function of the 2×2 determinantal ring k[X]/I₂(X) for arbitrary fields.
• To establish that this function is a polynomial in q, extending beyond asymptotic behavior.
• To derive closed-form expressions for the Hilbert-Kunz function and multiplicity when m ≤ 2.
• To provide a recursive computational framework using Gröbner bases applicable to other ideals beyond the maximal one.
• To offer a constructive method that computes the full function, not just the leading multiplicity, unlike prior combinatorial approaches.

Proposed method

• Uses a diagonal monomial order on k[X] to define a Gröbner basis for the ideal I₂(X) generated by 2×2 minors of a generic matrix.
• Introduces the concept of 'staircase monomials' and 'stair monomials' to characterize standard monomials modulo I₂(X).
• Develops a recursive formulation for the length λ(k[X]/(I₂(X) + m^[q])) by counting monomials in specific regions defined by the staircase structure.
• Applies binomial identities and summation formulas (e.g., Lemma A.1–A.4) to simplify complex sums arising from the recursive counting.
• Leverages the structure of the determinantal ring as a quotient to reduce the computation to counting monomials in a lattice with constraints.
• Uses the recursive length formula to prove that the Hilbert-Kunz function is a polynomial in q, with explicit degree m+n−1.

Experimental results

Research questions

• RQ1 Is the generalized Hilbert-Kunz function of the 2×2 determinantal ring k[X]/I₂(X) a polynomial in q for arbitrary fields and m≤2?
• RQ2 Can a recursive Gröbner basis-based method compute the full Hilbert-Kunz function, not just the multiplicity?
• RQ3 What closed-form expressions exist for the Hilbert-Kunz function and multiplicity when m=2 and n≥2?
• RQ4 How does the Hilbert-Kunz function behave across different characteristics, and is it independent of the field?
• RQ5 Can the recursive method be extended to compute lengths for other ideals beyond the maximal one?

Key findings

• The generalized Hilbert-Kunz function for 2×2 determinantal rings is proven to be a polynomial in q, with degree m+n−1.
• For m≤2, the Hilbert-Kunz function admits a closed-form expression independent of the field’s characteristic.
• The Hilbert-Kunz multiplicity for m=2 is given by e_HK = (n! / (n-1)!) S(n-1, n) - (1/(n-1)!) ∑_{r=1}^{1} ∑_{s=1}^{2-r} binom(2,r+s) binom(n,s) (-1)^{2+r} s^{n-1}, simplifying to a rational function.
• The recursive method enables computation of the full function, not just the leading term, providing exact values for all q.
• The method generalizes to other ideals, as demonstrated by the recursive counting of monomials in constrained regions.
• The approach confirms that the Hilbert-Kunz function is polynomial, resolving a key open question for this class of rings.