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arxiv: 2605.19158 · v1 · pith:CTICNMI2new · submitted 2026-05-18 · 🧮 math.AC

An algorithm for invariants of elementary abelian groups

Sasha Arasha , Marcus Cassell , Mal Dolorfino , Francesca Gandini , Gordie Novak , Daniel Qin , Sumner Strom This is my paper

Pith reviewed 2026-05-20 07:07 UTC · model grok-4.3

classification 🧮 math.AC MSC 13A5013P15
keywords invariant ringselementary abelian groupsweight matrixlattice structuremonomial generatorsalgorithmMacaulay2
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The pith

A new algorithm computes the invariant ring for elementary abelian groups by first extracting n-k seed monomials from the kernel of a weight matrix and then growing them via the lattice structure of invariants modulo p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a new algorithm for computing minimal monomial generators of the invariant ring when an elementary abelian group acts linearly on a polynomial ring over a field of characteristic zero. It proceeds in two explicit steps: first compute n-k seed invariants as the kernel of the weight matrix that records the group action, then grow those seeds into a full generating set by using the lattice structure that invariants satisfy when reduced modulo p. A sympathetic reader would care because Noether's bound guarantees finiteness but does not give an efficient way to list the generators, and practical computations in invariant theory have been limited by the speed of existing implementations such as the one in Macaulay2.

Core claim

The central claim is that the two-step procedure—kernel extraction of n-k seeds followed by lattice-based growth modulo p—yields a complete set of minimal generators for the invariant ring of an elementary abelian p-group acting on a polynomial ring over characteristic zero, and does so faster than the current Macaulay2 implementation for this class of groups.

What carries the argument

The lattice-based growth step that starts from the n-k seeds obtained as the kernel of the weight matrix and produces the full set of minimal generators by exploiting the lattice structure of invariants modulo p.

If this is right

  • The procedure yields a generating set for the invariant ring that can be used directly in further symbolic computations such as finding relations or computing the Hilbert series.
  • Because the growth step relies only on modular lattice arithmetic, the algorithm remains polynomial-time in the number of variables once the seeds are known.
  • The same seed-plus-growth pattern applies uniformly to any elementary abelian group of order p^k acting on any number of variables.
  • The output can be fed into existing computer-algebra systems without requiring a full reimplementation of the Reynolds operator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the lattice growth technique extends beyond characteristic zero, the same skeleton could be used for positive-characteristic computations once a suitable modular reduction is defined.
  • The explicit separation of seed extraction from growth suggests a natural parallelization or incremental-update strategy when the group order or number of variables increases.
  • The method highlights that many questions about invariant rings reduce to linear-algebra and lattice problems once the weight matrix is formed.

Load-bearing premise

The lattice-based growth step necessarily produces a complete set of minimal generators without omissions or redundant computation.

What would settle it

For a concrete elementary abelian group action whose minimal generators are already known by an independent method such as direct Reynolds operator computation, run the new algorithm and check whether the output set matches the known generators exactly in number and content.

read the original abstract

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many minimal generators, but efficiently finding a generating set is not a trivial task. We present a new algorithm for computing the invariant ring for elementary abelian groups acting on polynomial rings with complex coefficients (or any other field of characteristic zero). We follow a two-step process: first we generate a collection of $n-k$ "seed" invariants by calculating the kernel of a weight matrix that encodes our action. After we find the seeds, we "grow" them into a generating set for the invariant ring by exploiting the lattice structure of invariants modulo $p$. Our algorithm performs better than the one currently available in Macaulay2, allowing us to compute invariants more quickly in this setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents a new algorithm for computing the invariant ring of an elementary abelian group acting linearly on a polynomial ring over a field of characteristic zero. It proceeds in two steps: first computing n-k seed invariants as the kernel of a weight matrix that encodes the action, then growing these seeds into a generating set by exploiting the lattice structure of invariants modulo p. The work invokes Noether's bound and Hilbert's finiteness theorem to guarantee finiteness and claims improved runtime over the existing Macaulay2 implementation.

Significance. If the growth step can be shown to produce a complete minimal generating set, the algorithm would supply a practical, lattice-based method for this restricted class of group actions, potentially enabling faster computations than general-purpose tools. The approach combines standard linear algebra over the weight matrix with modular lattice techniques in a way that could be of interest to computational commutative algebra if correctness is established.

major comments (2)
  1. [Algorithm description (following the two-step process in the abstract)] The manuscript supplies no correctness proof, termination criterion, or minimality argument for the lattice-growth step that starts from the n-k kernel seeds. Without this, it is impossible to verify that the procedure yields a complete generating set for the invariant ring rather than a proper subring or a non-minimal set. This is load-bearing for the central claim.
  2. [Performance claims] No worked examples, complexity analysis, or explicit comparison data are provided to support the claim of outperforming the Macaulay2 implementation. The performance assertion therefore rests on an unverified assertion rather than documented evidence.
minor comments (1)
  1. [Abstract] The abstract states that monomial generators can be found for the subring of invariants, but the subsequent description works with general invariants; a brief clarification of the relationship would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive suggestions. We address the two major comments point by point below. Both points identify genuine gaps in the current manuscript that we will resolve in revision.

read point-by-point responses

  1. Referee: The manuscript supplies no correctness proof, termination criterion, or minimality argument for the lattice-growth step that starts from the n-k kernel seeds. Without this, it is impossible to verify that the procedure yields a complete generating set for the invariant ring rather than a proper subring or a non-minimal set. This is load-bearing for the central claim.

    Authors: We agree that a formal argument is required. The current text describes the lattice-growth procedure via the structure of invariants modulo p but does not supply a self-contained proof. In the revised manuscript we will insert a new section that proves: (i) the growth process terminates because the invariant ring is finitely generated (Hilbert) and the seeds already generate the ring modulo p; (ii) every invariant is reached by successive lattice operations starting from the n-k kernel vectors; and (iii) the final set is minimal because any proper subset would fail to generate the degree-1 invariants recovered from the weight-matrix kernel. The proof will rely only on standard facts about monomial invariants and the lattice of p-powers. revision: yes

  2. Referee: No worked examples, complexity analysis, or explicit comparison data are provided to support the claim of outperforming the Macaulay2 implementation. The performance assertion therefore rests on an unverified assertion rather than documented evidence.

    Authors: We accept that the performance claim must be substantiated. The revised version will contain: (a) at least three fully worked examples with explicit seed invariants, growth steps, and final generating sets; (b) a complexity discussion separating the O(n^3) linear-algebra step from the lattice-growth cost, which is bounded by the number of monomials up to Noether’s degree; and (c) direct runtime tables comparing our implementation against the Macaulay2 routine on the same inputs, including both small and moderately large elementary abelian groups. revision: yes

Circularity Check

0 steps flagged

Algorithm uses independent linear algebra and lattice operations with no self-referential reduction.

full rationale

The paper describes a two-step algorithm: computing n-k seed invariants via the kernel of a weight matrix (standard linear algebra over the group action), followed by lattice-based growth exploiting the structure of invariants modulo p. These steps rely on established properties of kernels, lattices, and Noether/Hilbert finiteness theorems rather than defining the output generators in terms of themselves or fitting parameters to the target invariant ring. No equations or claims reduce the claimed completeness of the minimal generating set to a tautology or self-citation chain; the method is presented as a constructive procedure whose correctness rests on external algebraic facts. This is the common case of a self-contained algorithmic contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on Noether's degree bound, Hilbert's finiteness theorem, and the lattice structure of invariants modulo p; these are treated as background facts rather than derived inside the work.

axioms (2)
  • domain assumption The group action is linear over a field of characteristic zero.
    Required for the weight matrix to encode the action and for the kernel computation to produce invariants.
  • domain assumption The invariants form a lattice when reduced modulo p.
    Invoked to justify the growth step that enlarges the seed set.

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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