Frobenius-Power Ideals and Hyperplane Avoidance for Representable Matroids

Amir Jafari

Frobenius-Power Ideals and Hyperplane Avoidance for Representable Matroids

T0 review · 0 major / 3 minor · reviewed 2026-08-28 · deepseek-v4-flash

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Pith's one-line read For matroids over F_{p^k}, extension degree at least the decomposition number forces a positive characteristic value.

desk verdict A clean, correct new theorem on characteristic polynomial positivity via Frobenius-power ideals; doesn't resolve the prime-field conjecture but deserves a serious look. read the letter →

arxiv 2608.23647 v1 pith:J665V5YL submitted 2026-08-24 math.CO

Amir Jafari This is my paper · ORCID

classification math.CO MSC 05B3511T0613F2052C35

keywords matroidcharacteristicpolynomialdecompositionnumberFrobenius-poweridealsnowhere-zerolinearmapsCombinatorialNullstellensatzfinite-fieldhyperplanearrangementsrepresentablematroidsprime-fieldavoidanceconjecture

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The pith

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

The reading

The paper proves a uniform positivity criterion for the characteristic polynomial of a representable matroid. If a finite matroid $M$ is representable over $\mathbb{F}_{p^k}$, and its ground set can be covered by $r$ independent sets, then $\chi_M(p^k)>0$ whenever $k\ge r$. The equivalent geometric statement is that in every spanning representation of $M$ by vectors in $\mathbb{F}_{p^k}^{\,n}$, some linear functional is nonzero on all representing vectors; the central hyperplanes of the representation cannot cover the dual space. The result matters because it gives a single condition, independent of the characteristic and of the number of independent sets, that guarantees a common nowhere-zero point for several invertible linear maps. The proof achieves this by storing exponent information in Frobenius-power ideals that are invariant under every linear change of coordinates.

What carries the argument

The load-bearing object is the Frobenius-power ideal $J_s=(X_1^{p^s},\dots,X_n^{p^s})$ inside the polynomial ring in $n$ variables over a field of characteristic $p$. Lemma 3.1 shows that every invertible linear change of coordinates preserves $J_s$, while a polynomial fails to lie in $J_s$ exactly when it contains a nonzero monomial whose exponent in each variable is smaller than $p^s$. This invariance is what lets the proof normalize several invertible linear systems in succession without losing the exponent bounds already obtained. Proposition 3.2, the iterated bounded-monomial lemma, shows that the product of the coordinate products of any $s$ invertible matrices contains such a bounded monomial; the coefficient form of the combinatorial Nullstellensatz then converts that monomial into an actual point of the finite-field grid where the product is nonzero, yielding the common nowhere-zero vector.

What would settle it

Testing all pairs of invertible $2\times 2$ matrices over $\mathbb{F}_4$ for a common nowhere-zero vector would settle the smallest nontrivial case of the matrix theorem; a single pair without such a vector would refute the central claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.1: for a finite matroid $M$ representable over $\mathbb{F}_q$ with $q=p^k$, writing $r=\mathrm{dec}(M)$ for the least number of independent sets covering the ground set and $n=\mathrm{rk}(M)$, if $k\ge r$ then $\chi_M(q)>0$. The theorem is proved through an equivalent matrix statement: any $r$ invertible $n\times n$ matrices over $\mathbb{F}_q$ with $k\ge r$ admit a vector $x$ such that every $A_ix$ has all coordinates nonzero. The matroid result follows by partitioning the ground set into $r$ independent sets, extending each to a basis, and applying the matrix statement to the resulting coordinate systems; the counting argument identifies the number of avoiding functionals with $\chi_M(q)$. The paper also delineates the limits of the theorem: projective geometries over a subfield give representable matroids with $\chi_M(q)=0$ when $k<r$, and without representability over the evaluation field no linear lower bound on $q/\mathrm{dec}(M)$ can force positivity.

Load-bearing premise

The argument stands on Lemma 3.1, the claim that the ideal generated by the $p^s$-th powers of the variables is sent to itself by every invertible linear change of coordinates; if this invariance failed, the exponent bounds obtained in one coordinate system could not be carried through later normalizations and the matrix theorem would collapse.

Editorial extensions

If this is right

The matrix theorem recovers the known two-map result over proper prime-power fields as the special case $r=2$, $k\ge 2$.
Every matroid representable over $\mathbb{F}_{p^k}$ whose decomposition number is at most $k$ has $\chi_M(q)>0$, so the characteristic value cannot vanish or be negative under that hypothesis.
Because the hypothesis is $k\ge r$ rather than a bound on the prime, the result is uniform: increasing the extension degree of the field forces avoidance regardless of the characteristic.
The projective-geometry construction produces $\mathbb{F}_{p^k}$-representable matroids with $k<r$ and $\chi_M(p^k)=0$, showing the inequality in the theorem cannot be weakened to $k<r$ in general; in many ranges $\chi_M(p^k)=0$ even when $q>2r-1$.
Without representability over the evaluation field, for every constant $C$ there are matroids with $q>C\cdot\mathrm{dec}(M)$ and $\chi_M(q)<0$, so the representability hypothesis is essential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Frobenius-ideal argument is not tied to matroids: the same coordinate-invariant storage of exponent bounds should apply to any avoidance problem encoded by several systems of linear forms, where the threshold $k\ge r$ in the number of systems is the natural uniform condition.
The examples in Section 6 leave open whether the bound $k\ge r$ is sharp; a plausible strengthening would be the existence of $\mathbb{F}_q$-representable matroids with $\mathrm{dec}(M)=k-1$ and $\chi_M(q)=0$, which the current constructions do not provide.
The prime-field conjecture attributed in the paper would, at $r=2$, imply the long-standing two-map conjecture over prime fields; the extension-field counterexamples show why the integer inequality $q>2r-1$ is not sufficient over proper extension fields, though they do not test the prime-field statement.
One testable extension is to ask whether the theorem holds with $k\ge r-1$ for simple matroids, since the obstruction examples all use a rank-one uniform direct sum to inflate the decomposition number.

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Formalized claims in Lean

Claim #1: On the paper's own terms, the central discovery is Theorem 2.1: for a finite matroid $M$ representable over $\mathbb{F}_q$ with $q=p^k$, writing $r=\mathrm{dec}(M)$ for the least number of independent sets covering the ground set and $n=\mathrm{rk}(M)$, if $k\ge r$ then $\chi_M(q)>0$. The theorem is proved through an equivalent matrix statement: any $r$ invertible $n\times n$ matrices over $\mathb

/-- @claim 1 On the paper's own terms, the central discovery is Theorem 2.1: for a finite matroid $M$ representable over $\mathbb{F}_q$ with $q=p^k$, writing $r=\mathrm{dec}(M)$ for the least number of independent sets covering the ground set and $n=\mathrm{rk}(M)$, if $k\ge r$ then $\chi_M(q)>0$. The theorem is proved through an equivalent matrix statement: any $r$ invertible $n\times n$ matrices over $\mathb -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing to know: this paper gives a clean sufficient condition for positivity of the characteristic polynomial of a matroid at the size of a finite field, via a new Frobenius-power ideal technique. It doesn't solve the big Alon-Jaeger-Tarsi conjecture, but it does something genuinely new and the proof checks out.

What's new is the idea of storing exponent bounds in an ideal J_s=(X_1^{p^s},...,X_n^{p^s}) that is GL_n-invariant in characteristic p, so that successive coordinate changes don't destroy the monomial information accumulated at earlier stages. That's a real trick, and it gives a uniform version for any number r of independent sets in a decomposition: if the field has size p^k with k>=r, then chi_M(q)>0. This complements the large-prime results of Nagy-Pach and Nagy-Pach-Tomon; it's uniform in characteristic and in r, rather than in the prime. Recovering the Alon-Tarsi two-map theorem as a special case is a nice sanity check. Section 6's counterexamples are also careful: projective geometries show the direct analogue over extension fields fails even when q>2r-1 in many cases, and Proposition 6.4 shows representability over the field of evaluation is essential. These sharply delimit the theorem.

Soft spots are minor. The paper doesn't claim sharpness of the k>=r threshold, so that's fine, but the reader should know the condition is sufficient, not necessary. Proposition 6.2's arithmetic checks out; I verified the ratio comparison and the direct-sum argument. The only real limitation is that the theorem says nothing about prime fields, and Conjecture 6.1 remains open; the paper is honest about that. The counterexamples target extension fields or non-representable matroids, so they don't indicate where a prime-field proof might fail. The paper is self-contained except for standard facts (Combinatorial Nullstellensatz, Edmonds's partition theorem, Oxley's formula), and the citations are appropriate. I found no circularity and no fitted parameters.

This is for matroid theorists, finite-field arrangement people, and anyone interested in Alon-Tarsi type questions. It deserves a serious referee. The theorem is new, the method is reusable, and the write-up is clear. The stress-test note agrees: no load-bearing gap. I'd accept this for peer review without hesitation.

Referee Report

0 major / 3 minor

Summary. The paper proves that for a finite matroid M representable over F_q with q=p^k (p prime), if dec(M) <= k, where dec(M) is the least number of independent sets covering the ground set, then the characteristic polynomial satisfies chi_M(q)>0. Equivalently, every spanning representation of M over F_q admits a linear functional nonzero on all represented vectors. The proof introduces the Frobenius-power ideals J_s=(X_1^{p^s},...,X_n^{p^s}), proves their invariance under GL_n, establishes an iterated bounded-monomial lemma (Proposition 3.2), and derives a simultaneous nowhere-zero statement for r invertible transformations via the coefficient form of the Combinatorial Nullstellensatz (Theorem 4.1). The passage from matrices to matroids uses a hyperplane-complement count (Lemma 5.1). The final section tests the scope of the theorem: it presents an unpublished prime-field conjecture, constructs projective-geometry counterexamples over proper extension fields for which chi=0 even when the numerical inequality q>2 dec(M)-1 holds, and shows that no linear q/dec(M) threshold can force positivity without representability over F_q.

Significance. The theorem is an elegant sufficient condition in terms of the extension degree rather than the prime, complementing the asymptotic results of Nagy, Pach, and Tomon and recovering the proper-prime-power case of the Alon-Tarsi theorem when dec(M)=2. The Frobenius-ideal technique is original, elementary, and self-contained, and the counterexamples in Section 6 are explicit and correctly delimit the theorem's scope. If the result stands, it is a valuable contribution to the study of characteristic polynomials of representable matroids and finite-field hyperplane arrangements.

minor comments (3)

[Title/Abstract] The phrase 'Hyperplane A voidance' contains an erroneous space and should read 'Hyperplane Avoidance'.
[Section 6, Conjecture 6.1] The attribution to an unpublished conjecture would benefit from a citation or a brief statement of where the conjecture can be verified.
[Proposition 6.2] The sentence 'The maximum is therefore attained by the whole projective geometry' is terse; adding one clause noting that the maximum over the possible ranks j is attained at j=k+1 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is self-contained and derives the theorem from independent lemmas.

full rationale

The paper's central derivation is not circular. Lemma 3.1 is proven directly from the Freshman's Dream identity in characteristic p: for each coordinate form ℓ_i(X)=Σ_j t_ij X_j, the paper shows ℓ_i(X)^{p^s}=Σ_j t_ij^{p^s} X_j^{p^s} ∈ J_s, and applying the same argument to T^{-1} gives equality. This is a self-contained proof, not an imported assumption. Proposition 3.2 is an induction whose base case is the explicit observation that Y_1⋯Y_n ∉ (Y_1^p,…,Y_n^p), and whose induction step multiplies by the monomial P_I(Y) to shift exponents by one; no conclusion of the theorem is assumed. Theorem 4.1 then applies the coefficient form of the Combinatorial Nullstellensatz to the homogeneous polynomial F of degree rn, with the exponent bound β_j+1 ≤ p^{r-1} < p^r ≤ p^k = q obtained from Proposition 3.2. This is a genuine external theorem applied to an independently constructed polynomial. Lemma 5.1 is a direct inclusion–exclusion count using the standard subset expansion of the characteristic polynomial, and the passage from independent-set partitions to invertible matrices uses only the definition of dec(M). Section 6's counterexamples use standard cited results: Oxley's characteristic polynomial formula for projective geometries and Edmonds's matroid partition theorem. No fitted parameters are renamed as predictions, no load-bearing self-citations appear, and no uniqueness theorem is imported from the author's prior work. The proof therefore does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof introduces the Frobenius-power ideals J_s as a mathematical construction, not as a postulated entity. All external results used are standard theorems cited in the paper. No free parameters are fitted; the theorem is a parameter-free derivation from stated hypotheses.

assumptions (5)

standard math The coefficient form of the Combinatorial Nullstellensatz is valid over finite fields.
Used in the proof of Theorem 4.1 to convert a nonzero monomial coefficient with exponents below the field size into a nonzero point.
standard math Edmonds's matroid partition theorem: dec(M) equals the maximum of ceil(|A|/rk(A)) over nonempty A.
Used in Proposition 6.2 to compute the decomposition number of projective geometries.
standard math The characteristic polynomial of the projective geometry PG(k,p) is the product over i=0..k of (t-p^i).
Quoted from Oxley's Matroid Theory [6] and used to produce zero values at t=p^k.
domain assumption The characteristic polynomial of a direct sum of matroids is the product of the characteristic polynomials, and the decomposition number is the maximum of the two.
Used in Proposition 6.2 to build matroids with arbitrary dec(M) and chi_M(q)=0.
domain assumption Restrictions of representable matroids over a field F are representable over F; hence a matroid containing the Fano matroid is not representable over fields of odd characteristic.
Used in Proposition 6.4 to show the constructed matroid is not representable over the evaluation field.

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Pith. "Pith review of Frobenius-Power Ideals and Hyperplane Avoidance for Representable Matroids." pith.science (2026). https://pith.science/paper/J665V5YL

@misc{pith2026260823647,
  author       = {Pith},
  title        = {Pith review of: Frobenius-Power Ideals and Hyperplane Avoidance for Representable Matroids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J665V5YL}},
  note         = {Machine review of arXiv:2608.23647}
}

abstract

Let $q=p^k$, where $p$ is prime, and let $M$ be a finite matroid representable over ${\Bbb{F}}_q$. Write $\chi_M(t)$ for its characteristic polynomial and $\mbox{decop}(M)$ for the least number of independent sets needed to cover its ground set. We prove that $\chi_M(q)>0$ whenever $k\ge\mbox{decop}(M)$. Geometrically, the central hyperplanes determined by any representation of $M$ fail to cover the dual of the ambient vector space. The proof rests on the Frobenius-power ideals $(X_1^{p^s},\ldots,X_n^{p^s})$, $s\ge1$. Each is preserved by every linear change of coordinates, while nonmembership records the existence of a monomial whose exponent in every variable is bounded. This permits successive normalizations of several invertible systems of linear forms without losing the exponent bounds already obtained. The coefficient form of the Combinatorial Nullstellensatz then produces a common nowhere-zero point. Finally, we test the scope of the theorem. M.~J.~Moghaddamzadeh's unpublished conjecture predicts a stronger statement over prime fields. Projective geometries show that its direct analogue fails over proper extension fields, even under the same numerical inequality.

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

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N. Alon, Combinatorial Nullstellensatz,Combinatorics, Probability and Computing8 (1999), no. 1, 7–29, doi:10.1017/S0963548398003411

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Nagy and P

J. Nagy and P. P. Pach, The Alon–Jaeger–Tarsi conjecture via group ring identities,J. Eur. Math. Soc.(2025), published online first, doi:10.4171/JEMS/1640

work page doi:10.4171/jems/1640 2025
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J. Nagy, P. P. Pach, and I. Tomon, Hyperplane covers of finite spaces and applications, Trans. Amer. Math. Soc.379 (2026), no. 1, 137–156, doi:10.1090/tran/9483

work page doi:10.1090/tran/9483 2026
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work page 2011

Pith tools

Reviewed August 28, 2026 · model on record in the stance chip above.

[1] [1]

Combinatorial nullstellensatz

N. Alon, Combinatorial Nullstellensatz,Combinatorics, Probability and Computing8 (1999), no. 1, 7–29, doi:10.1017/S0963548398003411

work page doi:10.1017/s0963548398003411 1999

[2] [2]

Alon and M

N. Alon and M. Tarsi, A nowhere-zero point in linear mappings,Combinatorica9 (1989), no. 4, 393–395, doi:10.1007/BF02125351

work page doi:10.1007/bf02125351 1989

[3] [3]

Edmonds, Minimum partition of a matroid into independent subsets,J

J. Edmonds, Minimum partition of a matroid into independent subsets,J. Res. Nat. Bur. Standards Sect. B69B (1965), 67–72, doi:10.6028/jres.069B.004

work page doi:10.6028/jres.069b.004 1965

[4] [4]

Nagy and P

J. Nagy and P. P. Pach, The Alon–Jaeger–Tarsi conjecture via group ring identities,J. Eur. Math. Soc.(2025), published online first, doi:10.4171/JEMS/1640

work page doi:10.4171/jems/1640 2025

[5] [5]

J. Nagy, P. P. Pach, and I. Tomon, Hyperplane covers of finite spaces and applications, Trans. Amer. Math. Soc.379 (2026), no. 1, 137–156, doi:10.1090/tran/9483

work page doi:10.1090/tran/9483 2026

[6] [6]

Oxley,Matroid Theory, second edition, Oxford University Press, 2011

J. Oxley,Matroid Theory, second edition, Oxford University Press, 2011. 8

work page 2011