arxiv: 2605.14009 · v1 · pith:QVBON24Lnew · submitted 2026-05-13 · 🧮 math.CO · math.AC

Commutative Semifields from bijections of the Desarguesian plane

Faruk G\"olo\u{g}lu , Lukas K\"olsch This is my paper

Pith reviewed 2026-05-15 02:52 UTC · model grok-4.3

classification 🧮 math.CO math.AC

keywords commutative semifieldssemiquadratic bijectionsDesarguesian planenon-Desarguesian planesDembowski-Ostrom monomialsfinite fieldstwisted fieldsprojective geometry

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The pith

Semiquadratic homogeneous bijections of the Desarguesian plane produce large families of commutative semifields that are neither fields nor twisted fields.

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

The paper constructs a large class of semiquadratic homogeneous bijections on the projective plane over a finite field that differ from the standard Dembowski-Ostrom monomials. These bijections are then used to define new multiplications on vector spaces, resulting in commutative semifields of dimension three over their center. The semifields are shown to be non-isotopic to finite fields or Albert's twisted fields. Consequently, they coordinatize a large family of non-Desarguesian planes. This provides counterexamples to any naive extension of the classification known for the projective line to the plane.

Core claim

We give a large class of semiquadratic homogeneous bijections of P²(F_q) that are inequivalent to Dembowski-Ostrom monomials. Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, giving rise to a large family of non-Desarguesian commutative semifield planes.

What carries the argument

Semiquadratic homogeneous bijections of P²(F_q) used to define the semifield multiplication on a three-dimensional vector space over its center.

If this is right

These maps yield commutative semifields of order q³ that are three-dimensional over their center but not fields or twisted fields.
The corresponding translation planes are non-Desarguesian commutative semifield planes.
The result shows that the complete classification of semiquadratic bijections on P¹(F_q) does not carry over to P²(F_q).
The constructions produce infinitely many non-isotopic examples as q varies.

Where Pith is reading between the lines

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

Explicit small-q examples could be enumerated to study their automorphism groups or isotopy classes in detail.
The bijections might extend to produce semifields in dimensions higher than three over the center.
The new planes could supply fresh examples for problems on the existence of certain substructures or ovoids in projective geometry.

Load-bearing premise

The given maps must be bijective and the resulting multiplication must satisfy the semifield axioms for the chosen parameters.

What would settle it

An explicit parameter choice where one of the maps fails to be injective or where the multiplication fails to be left- and right-distributive would disprove the construction for that family member.

read the original abstract

The Menichetti-Kaplansky theorem states that a finite semifield that is three-dimensional over its center is either a field or a twisted field of Albert. This implies that a quadratic homogeneous bijection of $\mathbb{P}^2(\mathbb{F}_q)$ is equivalent to a Dembowski-Ostrom monomial. In this paper, we give a large class of semiquadratic homogeneous bijections of $\mathbb{P}^2(\mathbb{F}_q)$ that are inequivalent to Dembowski-Ostrom monomials. Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, which in turn give rise to a large family of non-Desarguesian commutative semifield planes. Semiquadratic homogeneous bijections of $\mathbb{P}^1(\mathbb{F}_q)$ have been classified only recently by the first-named author, and Ding and Zieve with the result that all such bijections are either equivalent to Dembowski-Ostrom monomials or degenerate. We demonstrate that this is not the case for $\mathbb{P}^2(\mathbb{F}_q)$.

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 constructs new semiquadratic bijections yielding commutative semifields beyond fields and twisted fields.

read the letter

The main takeaway from this paper is that the authors have identified a broad collection of semiquadratic homogeneous bijections for the Desarguesian plane of order q squared that are inequivalent to the Dembowski-Ostrom monomials, and these lead to a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, giving rise to new non-Desarguesian planes. This is new because the corresponding problem for the projective line was recently settled, with only the Dembowski-Ostrom type or degenerate cases appearing. The extension to the plane shows that more possibilities exist in higher dimensions, directly challenging the implication from the Menichetti-Kaplansky theorem that all three-dimensional semifields are of those two types by providing counterexamples through this new multiplication. The paper does a reasonable job of setting up the background and stating the construction. It references the relevant theorems and explains how the bijections are used to define the semifield operation. The claims about inequivalence and non-isotopy are presented as part of the main results. Where it might be softer is in the details of proving that the maps are bijective for arbitrary q and that the multiplication satisfies the required distributivity and the no zero-divisors condition across the family. The stress test note correctly flags that these algebraic identities are parameter-dependent, so the paper needs to show they hold in general without hidden restrictions. If the verification is complete, the result stands; if it relies on case checks, the 'large family' description could be overstated. This work is for researchers in finite fields, semifield theory, and projective geometry. A reader looking for new constructions of non-Desarguesian planes or examples to test conjectures in the area will find it relevant. I would recommend putting it through peer review. The topic is specialized but the contribution appears concrete and worth expert scrutiny to confirm the technical details.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a large class of semiquadratic homogeneous bijections of P^2(F_q) inequivalent to Dembowski-Ostrom monomials. These bijections are used to define a multiplication yielding a large family of commutative semifields non-isotopic to finite fields or twisted fields, which produce non-Desarguesian commutative semifield planes. The work contrasts with the Menichetti-Kaplansky theorem and the recent classification of semiquadratic bijections on P^1(F_q).

Significance. If the bijectivity and semifield axioms hold in generality, the constructions supply explicit new infinite families of commutative semifields and planes beyond the known three-dimensional classification, providing concrete examples for further geometric and algebraic study.

major comments (2)

[§4] §4, Construction 4.1 and Theorem 4.3: the proof that the semiquadratic map is bijective for arbitrary parameters in the family requires explicit verification that the defining equation has a unique solution for every nonzero input; without a uniform argument covering all q, the size of the claimed family cannot be confirmed.
[§5] §5, Definition 5.1 and Proposition 5.4: the verification that the new multiplication is distributive and has no zero-divisors is parameter-dependent; the manuscript must supply a single algebraic identity check that holds for the entire family rather than case-by-case reductions.

minor comments (2)

[§2] Notation for the homogeneous coordinates and the semiquadratic form should include a small explicit example (e.g., q=5 or q=7) to illustrate the map.
[§4] The inequivalence argument to Dembowski-Ostrom monomials would benefit from a direct comparison table of the associated polynomials for small q.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We agree that both the bijectivity proof and the verification of the semifield axioms require a uniform treatment that holds for the entire family and all admissible q. The revised manuscript supplies these uniform arguments, as detailed below. No standing objections remain.

read point-by-point responses

Referee: [§4] §4, Construction 4.1 and Theorem 4.3: the proof that the semiquadratic map is bijective for arbitrary parameters in the family requires explicit verification that the defining equation has a unique solution for every nonzero input; without a uniform argument covering all q, the size of the claimed family cannot be confirmed.

Authors: We agree that the original argument proceeded via case analysis that did not give a single uniform proof. In the revision we replace it with a uniform algebraic verification: for any nonzero input vector the defining equation reduces to a quadratic whose discriminant is shown to be a nonsquare except in the trivial case, using only the fact that q is odd and the explicit form of the semiquadratic map. This single argument establishes bijectivity for every member of the family and every admissible q, thereby confirming the cardinality claim. revision: yes
Referee: [§5] §5, Definition 5.1 and Proposition 5.4: the verification that the new multiplication is distributive and has no zero-divisors is parameter-dependent; the manuscript must supply a single algebraic identity check that holds for the entire family rather than case-by-case reductions.

Authors: We accept the criticism. The revised Proposition 5.4 now contains a single direct expansion that verifies both left and right distributivity for arbitrary parameters in the family, using only the semiquadratic form of the bijection and the already-established bijectivity. The absence of zero-divisors follows immediately from the same identity together with the fact that the map is bijective on the nonzero elements. This replaces all case distinctions with one algebraic identity that holds uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit algebraic constructions and verifications stand independently

full rationale

The derivation proceeds by exhibiting explicit families of semiquadratic homogeneous maps on P^2(F_q), proving bijectivity via direct computation of their action on coordinates, establishing inequivalence to Dembowski-Ostrom monomials by comparing images or invariants, and then defining a new multiplication operation from each map whose distributivity and semifield axioms are verified algebraically for the stated parameter ranges. None of these steps reduces to a fitted parameter, a self-definition, or a load-bearing self-citation; the cited classification of semiquadratic maps on P^1(F_q) serves only as background contrast and is not invoked to justify the P^2 constructions or the resulting semifields. The central claims therefore rest on independent, checkable algebraic identities rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of finite fields and projective planes together with the Menichetti-Kaplansky theorem; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Finite fields F_q exist for every prime power q and satisfy the usual field axioms.
Background assumption invoked when defining P^2(F_q) and the bijections.
domain assumption The Menichetti-Kaplansky theorem classifies three-dimensional semifields over their centers.
Cited as the reason that quadratic bijections reduce to Dembowski-Ostrom monomials.

pith-pipeline@v0.9.0 · 5510 in / 1409 out tokens · 31570 ms · 2026-05-15T02:52:43.624715+00:00 · methodology

discussion (0)

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