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arxiv: 1906.10560 · v1 · pith:6YDKSJI2new · submitted 2019-06-25 · 🧮 math.RT · math.AG

The generating rank of a polar Grassmannian

Ilaria Cardinali , Luca Giuzzi , Antonio Pasini This is my paper

Pith reviewed 2026-05-25 16:03 UTC · model grok-4.3

classification 🧮 math.RT math.AG
keywords polar Grassmanniansgenerating rankHermitian formsquadratic formsWitt indexcommutative division ringsfinite fieldsincidence geometry
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The pith

The generating rank of k-polar Grassmannians over commutative division rings is computed explicitly in new cases.

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

This paper determines the smallest size of a generating set for k-polar Grassmannians when the underlying division ring is commutative. The computation covers polar Grassmannians from both Hermitian and quadratic forms under stated dimension and Witt index constraints. A sympathetic reader would care because the generating rank fixes the minimal number of elements needed to produce the entire incidence structure. New explicit results are given for Hermitian forms when the vector space dimension exceeds twice the Witt index, and for certain quadratic forms over small finite fields the structures are shown to be generated already over the prime subfield once the ambient dimension passes six.

Core claim

The generating rank of k-polar Grassmannians defined over commutative division rings is computed. Among the new results, the generating rank is determined for k-Grassmannians arising from Hermitian forms of Witt index n on vector spaces of dimension N greater than 2n. For 2-Grassmannians arising from quadratic forms of Witt index n over vector spaces of dimension N between 2n and 2n plus 2 over the fields with 4, 8 or 9 elements, the paper proves that when N exceeds 6 the geometry can be generated over the prime subfield, which fixes the generating rank.

What carries the argument

The generating rank: the smallest cardinality of a set of points that generates the entire polar Grassmannian under the incidence relations.

If this is right

  • Explicit values for the generating rank are now available for all Hermitian polar Grassmannians with ambient dimension strictly larger than twice the Witt index.
  • For the listed quadratic cases with N greater than 6 the generating rank equals the dimension of the underlying vector space over the prime field.
  • Generating sets of the computed minimal size can be exhibited for each of the geometries treated.
  • The results enlarge the list of polar spaces whose generating rank is known beyond previously settled low-rank or low-dimension cases.

Where Pith is reading between the lines

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

  • The same generation-over-prime-subfield phenomenon may hold for quadratic forms over additional finite fields once the ambient dimension is large enough.
  • The computed ranks could be used to bound the minimal number of generators in related point-line geometries such as other buildings or Grassmannians.
  • Relaxing commutativity of the division ring would likely require separate arguments and could produce different ranks.

Load-bearing premise

The division rings must be commutative and the forms must be non-degenerate with the given Witt indices and the stated bounds on ambient dimension.

What would settle it

A single counterexample would be any k-polar Grassmannian over a commutative division ring whose smallest generating set is strictly larger than the size computed in the paper, or a quadratic 2-Grassmannian over F_4 with N greater than 6 that cannot be generated by elements defined over the prime field.

read the original abstract

In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector spaces of dimension $N > 2n$. We also study generating sets for the $2$-Grassmannians arising from quadratic forms of Witt index $n$ defined over $V(N,{\mathbb F}_q)$ for $q=4,8,9$ and $2n \leq N \leq 2n+2$. We prove that for $N >6$ they can be generated over the prime subfield, thus determining their generating rank.

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

0 major / 2 minor

Summary. The paper computes the generating rank of k-polar Grassmannians defined over commutative division rings. New results include explicit values for k-Grassmannians arising from Hermitian forms of Witt index n on vector spaces of dimension N > 2n. It also examines generating sets for 2-Grassmannians from quadratic forms of Witt index n over V(N, F_q) with q = 4,8,9 and 2n ≤ N ≤ 2n+2, proving that for N > 6 these can be generated over the prime subfield and thereby determining the ranks.

Significance. If the stated proofs and constructions hold, the results advance the study of polar geometries by supplying previously unknown explicit generating ranks for Hermitian cases with N > 2n and for selected quadratic cases over small finite fields. The direct constructions of generating sets and the extension to commutative division rings constitute concrete, verifiable contributions to the literature on Grassmannians and forms.

minor comments (2)
  1. The introduction would benefit from a brief recall of the definition of generating rank to improve accessibility for readers outside the immediate subfield.
  2. Notation for the underlying division rings and the precise statement of the dimension constraints (N > 2n) could be made uniform across theorems to avoid minor ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript on the generating ranks of k-polar Grassmannians and for recommending minor revision. The referee's summary accurately captures the scope of our results, including the new computations for Hermitian forms with N > 2n and the quadratic form cases over small finite fields.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives generating ranks for k-polar Grassmannians via explicit geometric constructions and proofs based on the properties of non-degenerate forms (Hermitian or quadratic) over commutative division rings, with stated Witt indices and dimension constraints. These computations rely on direct arguments about generating sets within the given vector space dimensions and field characteristics, without any reduction of the final rank values to self-defined quantities, fitted parameters from the same work, or load-bearing self-citations. The central claims are self-contained within the paper's scope and do not invoke uniqueness theorems or ansatzes from prior author work as the sole justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Computations rest on the standard axiomatic theory of polar spaces and Grassmannians; no free parameters, no new entities, and no ad-hoc constants are introduced in the abstract.

axioms (1)
  • domain assumption Polar spaces and their Grassmannians are defined over commutative division rings with the usual incidence and form axioms.
    The paper invokes the established framework of forms on vector spaces to define the objects whose generating ranks are computed.

pith-pipeline@v0.9.0 · 5650 in / 1188 out tokens · 44176 ms · 2026-05-25T16:03:29.781456+00:00 · methodology

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Reference graph

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