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Polar degrees of the coordinate-wise inverse of a linear subspace are the coefficients of a substitution of the reduced characteristic polynomial of its matroid.

2026-06-26 01:27 UTC pith:BY4IRASM

load-bearing objection The paper gives a direct substitution formula for polar degrees of reciprocal linear spaces in terms of the reduced characteristic polynomial and settles two open conjectures on matroid discriminants.

arxiv 2606.26281 v1 pith:BY4IRASM submitted 2026-06-24 math.CO math.AG

Polar Degrees of Matroids

Clara Briand , Leonie Kayser , Julian Weigert This is my paper
classification math.CO math.AG
keywords matroidspolar degreesreciprocal linear spacesconormal varietiescharacteristic polynomialmatroid discriminants
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that polar degrees attached to the reciprocal of a linear subspace L in projective space arise directly as coefficients after a substitution into the reduced characteristic polynomial of the matroid M(L). The argument proceeds by identifying the relevant conormal varieties with the combinatorial conormal fan of the matroid, so that geometric degree data transfers to polynomial coefficients. This supplies an explicit combinatorial formula and, as a corollary, confirms two previously open conjectures on matroid discriminants.

Core claim

We show that the polar degrees of the coordinate-wise inverse of a linear subspace L ⊆ ℙ^n are given by the coefficients of a substitution of the reduced characteristic polynomial of the associated matroid M(L). Our proof connects the geometry of conormal varieties of reciprocal linear spaces to the combinatorial conormal fan of M(L). As a corollary, we settle two open conjectures regarding matroid discriminants.

What carries the argument

The combinatorial conormal fan of M(L), which encodes the conormal varieties of the reciprocal linear space so that polar degrees become coefficients of the substituted reduced characteristic polynomial.

Load-bearing premise

The geometry of the conormal varieties of reciprocal linear spaces aligns with the combinatorial conormal fan of M(L) so that polar degrees transfer directly to coefficients of the substituted polynomial.

What would settle it

Pick a concrete matroid such as the uniform matroid of rank 2 on 4 elements, compute its polar degrees by direct resolution of the corresponding reciprocal linear space, and check whether those numbers equal the coefficients obtained from the prescribed substitution into its reduced characteristic polynomial.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Polar degrees of reciprocal linear spaces reduce to a substitution into a single matroid polynomial.
  • Any matroid invariant that can be read from the reduced characteristic polynomial now yields explicit polar-degree formulas.
  • The two settled conjectures supply combinatorial expressions for the matroid discriminants in question.

Where Pith is reading between the lines

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

  • The same fan correspondence may let other algebraic invariants of reciprocal spaces be read off matroid polynomials.
  • Algorithms that enumerate matroid bases or flats could now compute polar degrees without solving systems of equations.
  • The result suggests testing whether further degree sequences attached to linear spaces admit similar matroid substitutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper proves that the polar degrees of the coordinate-wise inverse of a linear subspace L ⊆ ℙ^n are the coefficients obtained by a specific substitution into the reduced characteristic polynomial of the associated matroid M(L). The argument identifies the conormal variety of the reciprocal linear space with the combinatorial conormal fan of M(L) via explicit coordinate-wise inversion on Plücker coordinates, shows that polar-degree extraction commutes with this identification, and extracts the degrees from the fan's Hilbert series after substitution. Two open conjectures on matroid discriminants follow as special cases.

Significance. If the identification holds, the result supplies an explicit, parameter-free combinatorial formula for the polar degrees of reciprocal linear spaces and resolves two conjectures on matroid discriminants as direct corollaries. The geometric-combinatorial bridge via conormal fans is a substantive contribution; the derivation relies only on standard matroid axioms with no ad-hoc parameters, boundedness assumptions, or post-hoc fitting.

minor comments (2)
  1. [§2] §2: the precise substitution rule into the reduced characteristic polynomial is stated after the fan isomorphism; moving the substitution formula to the statement of Theorem 1.1 would improve readability.
  2. [Introduction] The proof sketch in the introduction refers to 'the indicated substitution' without an equation number; adding an explicit displayed equation for the substitution would help readers trace the degree extraction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the main theorem, and recommendation to accept. The assessment correctly identifies the combinatorial formula for polar degrees and the resolution of the two conjectures on matroid discriminants as corollaries.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central result equates polar degrees of reciprocal linear spaces to substituted coefficients of the reduced characteristic polynomial via an explicit geometric identification of the conormal variety with the matroid's combinatorial conormal fan. This proceeds from coordinate-wise inversion on Plücker coordinates, commutation of polar degree extraction, and standard matroid axioms, without any reduction to fitted inputs, self-definitional loops, or load-bearing self-citations. The two settled conjectures are direct corollaries. No enumerated circularity pattern applies.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract introduces no new free parameters or invented entities; the result relies on standard associations between linear subspaces and matroids plus existing notions of conormal varieties and fans.

axioms (1)
  • domain assumption Linear subspaces correspond to matroids via linear dependence relations
    Standard construction in matroid theory.

pith-pipeline@v0.9.1-grok · 5589 in / 1134 out tokens · 28468 ms · 2026-06-26T01:27:15.422294+00:00 · methodology

0 comments
read the original abstract

We show that the polar degrees of the coordinate-wise inverse of a linear subspace $L \subseteq \mathbb{P}^n$ are given by the coefficients of a substitution of the reduced characteristic polynomial of the associated matroid $\mathrm{M}(L)$. Our proof connects the geometry of conormal varieties of reciprocal linear spaces to the combinatorial conormal fan of $\mathrm{M}(L)$. As a corollary, we settle two open conjectures regarding matroid discriminants.

Figures

Figures reproduced from arXiv: 2606.26281 by Clara Briand, Julian Weigert, Leonie Kayser.

Figure 1
Figure 1. Figure 1: Lattice of flats of Mex. Highlighted are a flag of flats (red) and a decreasing flag of flats (blue), see Definition 3.3. The size of such a collection is called the length of F and denoted by |F| := k. When working with loopless matroids we usually assume the elements of a flag of flats to be non-empty. As the terminology suggests, matroids are inspired by linear independence in vector spaces. We now expl… view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · 2 internal anchors

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    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978- 2012