Mathieu-Zhao spaces of polynomial rings

Arno van den Essen; Loes van Hove

arxiv: 1907.06106 · v1 · pith:C5XKXN76new · submitted 2019-07-13 · 🧮 math.AC · math.RA

Mathieu-Zhao spaces of polynomial rings

Arno van den Essen , Loes van Hove This is my paper

Pith reviewed 2026-05-24 21:59 UTC · model grok-4.3

classification 🧮 math.AC math.RA

keywords Mathieu-Zhao spacespolynomial ringsfinite codimension idealsclassificationdecision algorithmalgebraically closed fields

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The pith

Mathieu-Zhao spaces of k[x1,...,xn] that contain a finite-codimension ideal are completely described when k is algebraically closed of characteristic zero.

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

The paper classifies every Mathieu-Zhao space inside the polynomial ring k[x_1,...,x_n] that contains some ideal of finite codimension. It supplies an explicit description of all such spaces together with an algorithm that decides membership for any subspace of the concrete form I plus a finite-dimensional vector space when I itself has finite codimension. A reader would care because these spaces arise when studying derivations on polynomial rings, and a full list plus a decision procedure turns an abstract class into something that can be checked on concrete examples.

Core claim

We describe all Mathieu-Zhao spaces of k[x_1,…,x_n] which contain an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form I+kv_1+⋯+kv_r is a Mathieu-Zhao space, in case the ideal I has finite codimension.

What carries the argument

Mathieu-Zhao space: the named class of subspaces whose complete list, among those containing a finite-codimension ideal, is given by the paper's description.

If this is right

Every Mathieu-Zhao space containing a finite-codimension ideal must match one of the forms listed in the classification.
The supplied algorithm terminates and returns the correct yes/no answer for any candidate subspace built from a finite-codimension ideal plus finitely many extra polynomials.
The results apply uniformly for every number of variables n.

Where Pith is reading between the lines

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

The classification reduces recognition of Mathieu-Zhao spaces in this subclass to a finite check once an ideal of finite codimension is identified.
The algorithm can be implemented directly in any computer algebra system that handles polynomial ideals of finite codimension.
Similar decision procedures might be sought for Mathieu-Zhao spaces in other rings where finite-codimension ideals exist.

Load-bearing premise

The definition of a Mathieu-Zhao space together with the hypothesis that k is algebraically closed of characteristic zero.

What would settle it

An explicit subspace containing a finite-codimension ideal that satisfies the Mathieu-Zhao property yet falls outside the paper's described list, or an input to the algorithm that the procedure accepts or rejects contrary to the actual property.

read the original abstract

We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form $I+kv_1+\cdots+kv_r$ is a Mathieu-Zhao space, in case the ideal $I$ has finite codimension.

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.

Desk Editor's Note private letter to a colleague

The paper classifies Mathieu-Zhao spaces containing finite-codimension ideals in polynomial rings over algebraically closed fields of char 0 and supplies a decision algorithm for finite extensions of such ideals.

read the letter

The paper classifies all Mathieu-Zhao spaces in k[x1,...,xn] that contain a finite-codimension ideal and gives an algorithm to decide membership for subspaces of the form I + kv1 + ... + kvr when I has finite codimension. Both results are stated for k algebraically closed of characteristic zero. This is the concrete new content: an explicit description plus a workable decision procedure that exploits the finite-dimensional quotient. The setup is internally consistent because the finite-codimension hypothesis reduces questions about the Mathieu-Zhao condition to linear algebra on a finite-dimensional space, which supports the claim of decidability. The paper supplies the definition of Mathieu-Zhao space and carries out the classification under the stated field hypotheses, so the results hold in the domain they claim. The main limitation is the finite-codimension restriction itself. It makes the problem tractable but also confines the classification to a proper subclass of all subspaces; nothing in the work suggests an immediate extension beyond that class. That is a scope choice rather than an error. The citation pattern is standard for the subfield and does not rely on unverified self-reference. This work is for the small group of people already studying Mathieu-Zhao spaces. A reader outside that niche gets little value. It is worth sending to peer review because the claims are specific, the reduction to finite dimension is clear, and referees can check the classification and algorithm directly against the definitions.

Referee Report

0 major / 0 minor

Summary. The paper claims to describe all Mathieu-Zhao spaces of the polynomial ring k[x_1,…,x_n] (k algebraically closed of characteristic zero) that contain an ideal of finite codimension, and to supply an algorithm deciding whether a subspace of the form I + k v_1 + ⋯ + k v_r is a Mathieu-Zhao space whenever I has finite codimension.

Significance. If the classification and algorithm are correct, the work supplies an explicit description together with a decision procedure for a natural class of subspaces in polynomial rings; the finite-codimension hypothesis makes the quotient finite-dimensional and thereby renders the decidability claim consistent with the setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. The referee's summary accurately captures the main results.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper supplies the definition of Mathieu-Zhao spaces in the full text and derives the classification of those containing a finite-codimension ideal together with a decision algorithm for subspaces of the indicated form. These results rest on standard algebraic properties of polynomial rings over algebraically closed fields of characteristic zero and the finite-dimensionality of the quotient; no equations, fitted parameters, or self-citations are shown to reduce the claimed statements to their own inputs by construction. The derivation is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the standing field hypotheses (algebraically closed, char 0) are background assumptions rather than new postulates introduced by the paper.

pith-pipeline@v0.9.0 · 5583 in / 1214 out tokens · 21526 ms · 2026-05-24T21:59:42.251782+00:00 · methodology

Mathieu-Zhao spaces of polynomial rings

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)