A note on irreducible slice algebraic sets

Anna Gori; Fabio Vlacci; Giulia Sarfatti

arxiv: 2601.21466 · v2 · pith:M3Y2QUOKnew · submitted 2026-01-29 · 🧮 math.AG

A note on irreducible slice algebraic sets

Anna Gori , Giulia Sarfatti , Fabio Vlacci This is my paper

Pith reviewed 2026-05-21 15:16 UTC · model grok-4.3

classification 🧮 math.AG

keywords quaternionic slice regular polynomialsright radical idealsquasi-prime idealssymmetrizationirreducible algebraic setsV_c(I)slice algebraic geometry

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The pith

If a right radical ideal in quaternionic slice regular polynomials is quasi-prime, its symmetrized zero set is irreducible.

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

The paper proves that if I is a right radical and quasi-prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization of the common zero set with commuting components is an irreducible algebraic set. This combines with the authors' prior results to give an equivalence: for radical ideals, the zero set V_c(I) is irreducible exactly when I is quasi-prime. A sympathetic reader cares because the result supplies an algebraic test for irreducibility that works in the non-commutative quaternionic setting.

Core claim

If I is a right radical and quasi prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization S_{V_c(I)} is an irreducible algebraic set, where V_c(I) is the set of common zeros with commuting components of polynomials in I. Combining this with results from the previous paper yields that for radical I, V_c(I) is irreducible if and only if I is quasi prime.

What carries the argument

The symmetrization S_{V_c(I)} of the commuting-component zero set, which converts the common zeros into an algebraic set whose irreducibility is tied directly to the quasi-primality of the generating ideal.

If this is right

Right radical quasi-prime ideals correspond to irreducible symmetrized algebraic sets.
For any radical ideal the zero set V_c(I) is irreducible precisely when the ideal is quasi-prime.
Irreducibility in the quaternionic slice setting can be checked via the quasi-prime property of the ideal.

Where Pith is reading between the lines

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

The equivalence may support a version of primary decomposition for slice regular polynomial ideals.
Similar symmetrization techniques could apply to algebraic sets over other non-commutative division algebras.
Algorithms that test quasi-primality might now be used to decide irreducibility of quaternion varieties.

Load-bearing premise

The results of the previous paper on the converse direction and on the correspondence between radical ideals and irreducible sets hold without additional restrictions on the quaternionic polynomial ring.

What would settle it

A concrete counter-example consisting of a right radical quasi-prime ideal I for which S_{V_c(I)} is reducible would show the claim is false.

read the original abstract

In this short note we prove that if $I$ is a right radical and quasi prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization $\mathbb S_{V_c(I)}$ is an irreducible algebraic set, where $V_c(I)$ is the set of common zeros with commuting components of polynomials in $I$. Combining this fact with the results proved in our previous paper [3], we obtain that for $I$ radical, $V_c(I)$ is irreducible if and only if $I$ is quasi prime.

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

This note proves the missing direction: right radical quasi-prime ideals yield irreducible symmetrized commuting zero sets, completing the equivalence with the authors' earlier paper.

read the letter

The core contribution is a direct proof that if I is a right radical and quasi-prime ideal in the slice regular quaternionic polynomials, then the symmetrized set S of its commuting zeros is irreducible. This pairs with the results in [3] to give the full statement that, for radical ideals, V_c(I) is irreducible exactly when I is quasi-prime. The argument stays within the definitions of right radical, quasi-prime, and the symmetrization map, so the new direction does not loop back on itself. The handling of non-commutativity through right ideals rather than two-sided ones fits the ring structure and avoids forcing commutative shortcuts. The note is short and focused, which is appropriate for this incremental step. The main limitation is its dependence on [3] for the converse and the basic radical-to-irreducible link; any gaps there would carry over, though the stress-test indicates the present direction is self-contained. No other obvious holes appear in the logic or the citation pattern. This is useful reading for specialists already working on slice regular functions or algebraic geometry over quaternions. A broader audience will find it too narrow. I would send it to peer review as a concise, technically sound addition that rounds out an existing line of results without overreaching.

Referee Report

0 major / 2 minor

Summary. The manuscript is a short note proving that if I is a right radical and quasi-prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization S_{V_c(I)} of the commuting zero set V_c(I) is an irreducible algebraic set. Combined with results from the authors' prior paper [3], this yields the characterization that, for radical ideals I, V_c(I) is irreducible if and only if I is quasi-prime.

Significance. If correct, the result completes a characterization of irreducibility for slice algebraic sets over quaternions in terms of ideal properties (right radical and quasi-prime), extending the correspondence between radical ideals and irreducible sets established in [3]. This provides a concrete link between algebraic and geometric notions in a non-commutative polynomial ring setting, which may support further work on quaternionic algebraic geometry.

minor comments (2)

The note would benefit from a brief recall of the definitions of 'right radical ideal' and 'quasi-prime ideal' (even if they appear in [3]), to make the argument self-contained for readers.
Clarify the precise construction of the symmetrization map S in the opening paragraphs, including how it acts on the commuting zero set V_c(I).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its significance in completing the characterization of irreducibility, and recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; new implication proved independently

full rationale

The note directly establishes the forward implication (right radical and quasi-prime ideal implies symmetrized variety irreducible) from the definitions of right radical ideals, quasi-prime ideals, and the symmetrization map on the commuting zero set, without reducing to any fitted inputs, self-definitions, or unverified ansatzes. The converse and full iff characterization are explicitly delegated to the authors' prior independent paper [3], which does not create a self-referential loop or load-bearing dependency within the current derivation. The argument structure remains self-contained against external algebraic definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Pure proof paper; no numerical fitting, no new postulated objects, and no ad-hoc parameters introduced. Relies on standard properties of quaternions and the ring of slice regular polynomials as background.

axioms (1)

domain assumption The ring of quaternionic slice regular polynomials admits well-defined notions of right radical and quasi-prime ideals that interact with zero sets in the expected way.
Invoked implicitly when stating the theorem for this ring; standard in the subfield but not re-proved here.

pith-pipeline@v0.9.0 · 5609 in / 1284 out tokens · 37125 ms · 2026-05-21T15:16:14.652131+00:00 · methodology

A note on irreducible slice algebraic sets

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)