[Paper Review] An algorithm for the HK function for disjoint-term trinomial hypersurfaces

Shyamashree Upadhyay|arXiv (Cornell University)|Apr 24, 2012

Advanced Numerical Analysis Techniques4 references3 citations

TL;DR

This paper presents a computational algorithm for determining the Hilbert-Kunz function of disjoint-term trinomial hypersurfaces over any field of positive characteristic. By reducing the problem to solving systems of linear equations and analyzing matrix rank conditions, the method identifies cases where the Hilbert-Kunz multiplicity may be irrational, suggesting a pathway to constructing such examples via combinatorial matrix analysis.

ABSTRACT

A `trinomial hypersurface' is a hypersurface that is defined by a single polynomial having 3 non-constant terms in it and no constant term. A `disjoint-term trinomial hypersurface' is a trinomial hypersurface whose defining polynomial has the property that any 2 distinct terms in it have GCD equal to 1. In this article, I provide an algorithm for computing the Hilbert-Kunz function for any disjoint-term trinomial hypersurface in general, over any field of arbitrary positive characteristic. However, I do not provide any formula for the Hilbert-Kunz function.

Motivation & Objective

• To develop a general algorithm for computing the Hilbert-Kunz function of disjoint-term trinomial hypersurfaces over any field of positive characteristic.
• To investigate whether the Hilbert-Kunz multiplicity can be irrational for such hypersurfaces.
• To reduce the computation of the Hilbert-Kunz function to solving systems of linear equations and analyzing matrix ranks.
• To identify combinatorial patterns in matrix systems that may signal irrationality in the Hilbert-Kunz multiplicity.
• To lay the groundwork for proving irrationality of Hilbert-Kunz multiplicities using trinomial hypersurfaces as a minimal test case.

Proposed method

• The algorithm uses the 'mutation' process to transform the Hilbert-Kunz computation into a system of linear equations over monomials.
• It introduces term-ordering and combinatorial invariants like $1_{\text{min},A}$, $2_{\text{min},A}$, $(-2)_{\text{max},A}$, and $(-3)_{\text{max},A}$ to classify monomials.
• The method reduces the problem to checking solvability of linear systems represented by matrices $\mathfrak{B}_{A,f}^{\text{tr}}$, with solvability depending on specific inequalities among invariants.
• For ambiguous cases, the algorithm requires computing the rank of large matrices derived from combinatorial tables (e.g., tables 5, 6, 8), which exhibit structured patterns.
• The approach leverages linear algebra techniques to compute the length $l(R / \mathfrak{m}^{(p^n)} + J)$, equivalent to the Hilbert-Kunz function.
• The algorithm is designed to be general and applicable to any disjoint-term trinomial hypersurface, regardless of the number of variables or characteristic.

Experimental results

Research questions

• RQ1 Can the Hilbert-Kunz multiplicity of a disjoint-term trinomial hypersurface be irrational, and if so, under what conditions?
• RQ2 What combinatorial or algebraic features in the defining polynomial lead to non-rational Hilbert-Kunz multiplicities?
• RQ3 How do matrix rank computations in the algorithm relate to the asymptotic behavior of the Hilbert-Kunz function?
• RQ4 Can the structure of the linear systems derived from mutation reveal periodic or rhythmic patterns in the function’s growth?
• RQ5 Is it sufficient to study trinomial hypersurfaces to detect irrational Hilbert-Kunz multiplicities, as suggested by Monsky’s 5-variable example?

Key findings

• The algorithm provides a systematic way to compute the Hilbert-Kunz function for any disjoint-term trinomial hypersurface over a field of positive characteristic.
• In certain monomial cases, the solvability of the associated linear system depends on complex inequalities between combinatorial invariants, making the outcome non-trivial.
• When the system $\mathfrak{B}_{A,f}^{\text{tr}}\mathfrak{Y}^{\text{tr}} = \mathfrak{e}^{\text{tr}}$ is not solvable due to non-zero right-hand side and zero last row, the monomial is not in $A_c + J$, indicating a contribution to the Hilbert-Kunz function.
• For ambiguous cases where $\max\{1_{\text{min},A}, 2_{\text{min},A}\} \leq (-3)_{\text{max},A} < 1_{\text{min},A} + 2_{\text{min},A} - 1$, the membership in $A_c + J$ depends on the rank of specific matrices.
• The algorithm identifies that matrix rank computations are required for certain configurations, and these computations are the main obstacle to deriving a closed-form formula.
• The author suspects that the irregular behavior of these matrix ranks may underlie irrational Hilbert-Kunz multiplicities, suggesting a new route to proving irrationality in such invariants.