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On norming systems of linear equations

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Pith's one-line read Weighted solution counts identify a linear system up to isomorphism.

desk verdict A substantial and convincing framework for norming linear systems; the central theorems hold up, but the paper should supply a few omitted proofs and reduce reliance on an unpublished reference before final acceptance. read the letter →

arxiv 2411.18389 v1 pith:DQ3W3NDR submitted 2024-11-27 math.CO math.NT

Seokjoon Cho , David Conlon , Joonkyung Lee , Jozef Skokan , Leo Versteegen This is my paper · ORCID
classification math.COmath.NT MSC 05C6511B3005C35
keywords norminglinearsystemsweaklyisomorphismtheoremvariable-transitivityforcinguniformitynormsoverfinitefieldsrank-twoclassification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks which systems of linear equations over the vector spaces $\mathbb{F}_q^n$ are norming: the average $t_L(f)$ of $f(x_1)\cdots f(x_k)$ over all solutions to the system should define a norm on functions. The paper's central result is an isomorphism theorem: if two systems $L$ and $M$ give the same value $t_L(f)=t_M(f)$ for every non-negative function $f$ on every $\mathbb{F}_q^n$, then $L$ and $M$ are the same system up to row operations and column permutations. Using this, the paper proves that every weakly norming system is variable-transitive, in the sense that deleting any variable leaves an isomorphic subsystem, and it classifies all weakly norming systems of rank at most two. It also proves that every weakly norming system is forcing, meaning that the constant function is the unique minimizer of $t_L$ at a fixed mean. The motivating examples are the uniformity norms of additive combinatorics, and the paper shows how norming hypergraphs generate further norming systems.

What carries the argument

The load-bearing object is the solution-count functional $t_L(f_1,\dots,f_k)=\mathbb{E}_{(x_1,\dots,x_k)\in \operatorname{Sol}(L)} f_1(x_1)\cdots f_k(x_k)$, with $t_L(f)=t_L(f,\dots,f)$. Its Fourier-inversion form, $t_L(f_1,\dots,f_k)=\sum_{\xi\in \widehat{G}^m}\prod_{j=1}^k \widehat{f_j}(\sum_i L_{ij}\xi_i)$, turns norming conditions into spectral inequalities and is what lets the paper tell systems apart. The isomorphism proof also relies on the symmetrised functional $\tau_L(f_1,\dots,f_k)=\sum_{\pi\in S_k} t_L(f_{\pi(1)},\dots,f_{\pi(k)})$, the sum over all permutations of the $k$ inputs, together with tensor-power constructions that amplify any difference between non-isomorphic systems until it is visible in $\tau_L$. The subdivision operation, which replaces each column $v$ by two columns $v$ and $-v$, preserves weak norming and norming and generates the rank-two classification.

What would settle it

A computer search over $3\times k$ systems over $\mathbb{F}_2$ could try to find a weakly norming system with two non-isomorphic variable-deleted subsystems; by Corollary 4.6 no such system exists, so one explicit example would refute the paper's variable-transitivity theorem.

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Extended reading notes

Core claim

The discovery is that the functional $t_L(f)$, the average of $f(x_1)\cdots f(x_k)$ over the solution set of $L$, is a complete fingerprint of the system on non-negative inputs: equality of $t_L$ and $t_M$ on all such functions forces $L$ and $M$ to be isomorphic. The proof passes through a symmetrised version $\tau_L$ acting on $k$-tuples of complex functions, then uses Fourier inversion on $\mathbb{F}_q^n$ and tensor powers to make non-isomorphic systems produce different values. From this fingerprint property the paper derives structural constraints on weakly norming systems: they must be translation invariant, in the sense that shifting all coordinates by the same group element preserves solutions; their girth, the smallest support of a non-zero row-space vector, must be even; and their minimum-support row vectors must be balanced sign vectors with equally many $a$ and $-a$ entries. The rank-two classification says that every weakly norming $2\times k$ system is one of three explicit families, obtained from equality systems by subdivision. Finally, weak norming is shown to imply forcing, and every norming system is shown to admit a conjugation assignment under which it defines a norm for complex-valued functions.

Load-bearing premise

Proposition 2.2, stated without proof, asserts that weak norming is equivalent to a Holder-type inequality for non-negative functions, and the paper uses that equivalence repeatedly to derive structural constraints; if the equivalence failed, several main conclusions would lack foundation.

Editorial extensions

If this is right

  • If $L$ is weakly norming, then deleting any one of its variables always leaves the same subsystem up to isomorphism; in particular, no column of a non-zero weakly norming system is zero.
  • Every weakly norming system of rank one is a single balanced equation with no zero coefficients, and every weakly norming system of rank two is isomorphic to one of three explicit families.
  • Every weakly norming system is forcing: whenever $t_L(f)=\mathbb{E}[f]^k$, the function $f$ must be constant.
  • Every real-norming system can be given a conjugation assignment so that the same functional defines a norm on complex-valued functions.
  • Every weakly norming hypergraph gives rise, by parametrising its solution set by sums of vertex variables, to a weakly norming linear system; for graphs the resulting system has rank $|E(H)|-|V(H)|+\kappa(H)$ and its minimal dependencies are exactly the even cycles.

Reading between the lines

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

  • The isomorphism theorem likely gives a finite certificate for isomorphism: it may suffice to check $t_L$ on indicator functions of small subsets of a fixed $\mathbb{F}_q^n$, although the paper's proof uses many more functions and does not state such a bound.
  • The rank-two classification suggests a generative grammar for weakly norming systems through subdivision and disjoint unions; if this extends, higher-rank systems would be built from degenerate equality systems, which would connect the arithmetic theory to the reflection-group picture for graph norms.
  • Because weakly norming systems are forcing, they are natural candidates for quasi-randomness tests: equality of $t_L$ with its random value could certify that a function is structured, a consequence the paper states only implicitly.
  • The hypergraph construction gives a pipeline from norming hypergraphs to norms on functions; combining it with the forcing property may yield new lower bounds for arithmetic removal or density increment arguments, though the paper does not pursue these applications.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 5 minor

Summary. The paper initiates a systematic study of norming and weakly norming systems of linear equations over F_q^n. For an m x k system L it defines the functional t_L(f)=E_{x in Sol(L)} prod_i f(x_i) and asks when |t_L(f)|^{1/k} or t_L(|f|)^{1/k} defines a norm. The main results are an isomorphism theorem (Theorem 4.1) saying that equality of t_L and t_M on all nonnegative functions implies L and M are isomorphic up to row operations and column permutations; the corollary that weakly norming systems are variable-transitive (Corollary 4.6); necessary conditions including translation invariance, the Sidorenko property, even girth, and the existence of Schatten vectors (Theorem 5.1); a complete classification of weakly norming systems of rank at most two (Theorem 5.8); a proof that weakly norming systems are forcing (Theorem 6.3); a construction of norming systems from norming hypergraphs (Section 7); and a proof that every real-norming system admits a conjugation assignment making it complex-norming (Theorem 8.1). The arguments combine Fourier analysis, tensor-power manipulations, and an adaptation of Sidorenko's graph technique.

Significance. If the central theorems are correct, this is a substantial contribution to additive combinatorics. Theorem 4.1 is a powerful uniqueness criterion with no direct arithmetic predecessor and underpins the variable-transitivity result; the rank-two classification is the first complete classification in this setting; and the forcing and complex-norming results transfer a substantial body of graph-norm theory to linear systems. The paper is clearly written and gives detailed proofs for the main line, with the Fourier and tensor-product arguments checked carefully. The principal weaknesses are the number of supporting statements whose proofs are deferred or sketched (Proposition 2.2, Theorem 4.8, Corollary 8.3), a density-scaling issue in the proof of Proposition 6.1, and one terse step in Theorem 5.8; all appear repairable within the manuscript's scope.

major comments (6)
  1. [Section 2, Proposition 2.2] Proposition 2.2 states that L is weakly semi-norming if and only if the Holder-type inequality t_L(f_1,...,f_k) <= prod_i ||f_i||_{r(L)} holds for all nonnegative functions, but its proof is omitted. This equivalence is used repeatedly in the rest of the paper (Corollary 3.2, Theorem 5.1, Theorem 6.3), so the omission is load-bearing. Since the reader is only referred to 'a similar argument' to Proposition 2.1, please include the proof or a precise reference; the nonnegative setting requires care because the usual absolute-value renormalisation must be replaced by positivity and monotonicity of t_L on nonnegative inputs.
  2. [Section 6, Proposition 6.1] The first sentence of the proof says 'We may assume that n>1 by extending f to a larger F_q^n through adding an extra zero to each point in the support.' If 'adding an extra zero' means defining f' by f'(x,0)=f(x) and f'=0 outside the embedded copy of F_q^n, then the hypothesis is not preserved: t_L(f') = q^{m-k} t_L(f) while E[f'] = q^{-1} E[f], so the sign of t_L(f') - E[f']^k can change. The reduction should instead use the constant extension f'(x,z)=f(x), which preserves both t_L and the expectation, or the proof should be modified accordingly.
  3. [Section 8, Corollary 8.3] Corollary 8.3, which asserts that s_L^{1/k} is a norm on F, is essential to the proof of Theorem 8.1, but its proof is omitted as 'verbatim the same' as [15, Lemma 5.5]. Given that Theorem 8.1 is a main result and the omitted proof concerns a weak decoration functional, the authors should include the argument rather than rely on an unstated matching.
  4. [Section 5, proof of Theorem 5.8] At the end of the proof of Theorem 5.8, the sentence 'implying that L does not have the Holder property' omits the comparison that makes the contradiction explicit. Starting from t_L(f) <= t_L(f_1,...,f_k) - 2 alpha^k, one still has to compare t_L(f_1,...,f_k) with the upper bound supplied by Proposition 2.2 for the same tuple; the authors should insert this final step so that the displayed strict gap is shown to contradict the Holder inequality.
  5. [Section 4, Theorem 4.8] The proof of Theorem 4.8 (components of a weakly norming system are isomorphic and norming) is omitted, with the text saying it follows from Corollary 3.3 'but is otherwise exactly the same as' the simplified proof of [10, Theorem 1.2] in [7, Lemma 2.4]. Since this theorem is stated as an application of the isomorphism theorem and as a structural statement about norming systems, a proof sketch or an explicit reduction to the cited lemma should be included.
  6. [Section 5, Corollary 5.3] In the proof of Corollary 5.3, the claim that if m >= 2 then 'by using row operations, one can clearly make a vector with smaller support' is not immediate and is needed for the rank-two classification. The claim is true, but it requires the argument that a subspace of dimension at least 2 over F_q contains a nonzero vector with a zero coordinate, obtained by taking a suitable linear combination of two independent row vectors; please supply this justification.
minor comments (5)
  1. [Example 7.6] The displayed matrix for L_H in the case a=b=3 is not a valid 4 x 9 matrix as printed: it appears to have too few entries and two identical rows. Please correct the typesetting or the matrix itself.
  2. [Section 7, Theorem 7.7] The proof invokes Theorem 5.1 for norming systems, although Theorem 5.1 is stated for weakly norming systems; please add a one-sentence justification that norming systems are weakly norming, for instance via Proposition 3.5 and Proposition 2.1.
  3. [Example 7.8 and references] The reliance on [13], a private communication, for the statement that every norming system is equivalent to some U_k norm makes that remark unverifiable; the authors should mark it as conditional or provide a public reference.
  4. [Throughout] There are several typographical errors, including 't o' in the abstract, a stray ')' in the displayed equation in Proposition 4.4, and the 'bracehtipupleft' artifacts in Section 2; a thorough proofreading pass is needed.
  5. [Example 5.9] In the parametrisation of Sol(L), the notation is somewhat compressed because the variables (x_1,...,x_6) appear in the first line and again in the description of the incidence structure; a concise statement of which variables correspond to which edges of the singleton-pair incidence graph would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the core isomorphism theorem and its applications are derived from first principles, with no fitted input renamed as a prediction and no self-citation chain forcing the conclusions.

full rationale

The central claim, Theorem 4.1, is proved inside the paper: Lemma 4.2 uses elementary random-sign and random-unit averaging to lift equality on nonnegative functions to complex-valued symmetrized functionals, and Proposition 4.4 uses the Fourier inversion formula (2) with indicator Fourier transforms; non-isomorphism forces the intersection im(L^t) cap im(M^t) to have dimension at most m-1, giving a strict count separation. No step assumes t_L = t_M or uses the desired conclusion as an input. The subsequent applications (variable-transitivity, Corollary 4.6, rank-two classification Theorem 5.8, forcing Theorem 6.3) chain from Theorem 4.1, Proposition 2.2, and direct Fourier computations. The paper does omit some supporting arguments: Proposition 2.2 says 'whose proof we omit' but is the weakly norming analogue of the fully proved Proposition 2.1; Theorem 4.8 omits a proof said to follow [7, Lemma 2.4]; Corollary 8.3 omits a proof said to be verbatim [15, Lemma 5.5]. These are presentation gaps, not circular reductions. The cited prior results by overlapping authors (Conlon-Lee, Lee-Sidorenko) are published theorems about graph norms or domination inequalities, parameter-free and not fitted to this paper's data, so they function as real external support rather than as the paper's own conclusions recycled as premises. No fitted parameter is relabelled as a prediction, and no ansatz is imported by citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. It relies on a set of published external theorems, one unpublished private communication, and one internally stated but unproved equivalence (Proposition 2.2). These are the main axioms on which the central claims rest.

assumptions (6)
  • domain assumption Equivalence of weak semi-norming and the Holder inequality for non-negative functions (Proposition 2.2, proof omitted).
    Stated without proof; used throughout Sections 3-6 to derive necessary conditions for weakly norming systems.
  • standard math Fox-Pham-Zhao Theorem 1.4 characterizing Sidorenko 1 x k systems.
    Used in Theorem 6.2 to show a Sidorenko single equation is a pair of +/- coefficients.
  • standard math Lee-Sidorenko Lemmas 2.2, 2.3, 5.5 on decoration functionals and convexity.
    Used in Section 8 to prove existence of a complex norming conjugation assignment.
  • domain assumption Hatami-Hatami-Lovett equivalence of norming system norms to U_k norms (private communication [13]).
    Unpublished result cited only as private communication; used in Example 7.8.
  • standard math Garbe-Hladky-Lee Theorem 1.2 and Conlon-Lee Lemma 2.4 on graph norm components.
    Used for Theorem 4.8 on components of weakly norming systems.
  • standard math Standard Fourier analysis on F_q^n (inversion, Plancherel, convolution).
    Foundation for the Fourier reformulation of t_L in Proposition 2.3 and subsequent proofs.

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Pith. "Pith review of On norming systems of linear equations." pith.science (2026). https://pith.science/paper/DQ3W3NDR

@misc{pith2026241118389,
  author       = {Pith},
  title        = {Pith review of: On norming systems of linear equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQ3W3NDR}},
  note         = {Machine review of arXiv:2411.18389}
}
abstract

A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two.

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Forward citations

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Reference graph

Works this paper leans on

21 extracted references · 13 canonical work pages · cited by 2 Pith papers

  1. [13]

    Private communication

    Hamed Hatami. Private communication. 2024

    work page 2024

  2. [1]

    On a common-extendable, non-Sidorenko l inear system

    Daniel Altman. On a common-extendable, non-Sidorenko l inear system. Comb. Theory , 3(3):Paper No. 5, 12, 2023

    work page 2023

  3. [2]

    An approxima te version of Sidorenko’s con- jecture

    David Conlon, Jacob Fox, and Benny Sudakov. An approxima te version of Sidorenko’s con- jecture. Geom. Funct. Anal. , 20(6):1354–1366, 2010. doi:10.1007/s00039-010-0097-0

    work page doi:10.1007/s00039-010-0097-0 2010

  4. [3]

    Weak quasi- randomness for uniform hypergraphs

    David Conlon, Hiˆ ep H` an, Yury Person, and Mathias Schac ht. Weak quasi- randomness for uniform hypergraphs. Random Structures Algorithms , 40(1):1–38, 2012. doi:10.1002/rsa.20389

    work page doi:10.1002/rsa.20389 2012

  5. [4]

    Some advances on Sidorenko’s conjecture.J

    David Conlon, Jeong Han Kim, Choongbum Lee, and Joonkyun g Lee. Some advances on Sidorenko’s conjecture.J. London Math. Soc., 98(3):593–608, 2018. doi:10.1112/jlms.12142

    doi:10.1112/jlms.12142 2018

  6. [5]

    Finite reflection groups and graph norms

    David Conlon and Joonkyung Lee. Finite reflection groups and graph norms. Adv. Math. , 315:130–165, 2017. doi:10.1016/j.aim.2017.05.009

    doi:10.1016/j.aim.2017.05.009 2017

  7. [6]

    Sidorenko’s conjecture for blow-ups

    David Conlon and Joonkyung Lee. Sidorenko’s conjecture for blow-ups. Discrete Anal. , 2021:Paper No. 2, 13 pp., 2021. doi:10.19086/da. 29

    doi:10.19086/da 2021

  8. [7]

    Domination inequalitie s and dominating graphs

    David Conlon and Joonkyung Lee. Domination inequalitie s and dominating graphs. Math. Proc. Cam. Phil. Soc. , pages 1–18, 2024. doi:10.1017/S0305004124000185

    work page doi:10.1017/s0305004124000185 2024

Show all 21 references
  1. [8]

    Sparse graph counting and Kelley–Meka bounds for binary systems

    Yuval Filmus, Hamed Hatami, Kaave Hosseini, and Esty Kel man. Sparse graph counting and Kelley–Meka bounds for binary systems. Preprint available at arXiv:2311.12248

    arXiv

  2. [9]

    Common and Sidor enko linear equations

    Jacob Fox, Huy Tuan Pham, and Yufei Zhao. Common and Sidor enko linear equations. Q. J. Math., 72(4):1223–1234, 2021. doi:10.1093/qmath/haaa068

    2021 doi

  3. [10]

    Two rem arks on graph norms

    Frederik Garbe, Jan Hladk´ y, and Joonkyung Lee. Two rem arks on graph norms. Discrete Comput. Geom. , 67(3):919–929, 2022. doi:10.1007/s00454-021-00280-w

    2022 doi

  4. [11]

    A new proof of Szemer´ edi’s theorem

    William Timothy Gowers. A new proof of Szemer´ edi’s theorem. Geom. Funct. Anal., 11(3):465– 588, 2001. doi:10.1007/s00039-001-0332-9

    2001 doi

  5. [12]

    Graph norms and Sidorenko’s conjecture

    Hamed Hatami. Graph norms and Sidorenko’s conjecture. Israel J. Math. , 175:125–150, 2010. doi:10.1007/s11856-010-0005-1

    2010 doi

  6. [14]

    Towards a characterization of Sidorenko systems

    Nina Kamˇ cev, Anita Liebenau, and Natasha Morrison. Towards a characterization of Sidorenko systems. Q. J. Math. , 74(3):957–974, 2023. doi:10.1093/qmath/haad013

    2023 doi

  7. [15]

    On graph norms f or complex-valued functions

    Joonkyung Lee and Alexander Sidorenko. On graph norms f or complex-valued functions. J. Lond. Math. Soc. , 106(2):1501–1538, 2022. doi:10.1112/jlms.12604

    2022 doi

  8. [16]

    Operations with structures.Acta Math

    L´ aszl´ o Lov´ asz. Operations with structures.Acta Math. Acad. Sci. Hungar. , 18:321–328, 1967. doi:10.1007/BF02280291

    1967 doi

  9. [17]

    Graph homomorphisms: Open problems

    L´ aszl´ o Lov´ asz. Graph homomorphisms: Open problems. Unpublished manuscript, 2008

    2008

  10. [18]

    L´ aszl´ o Lov´ asz.Large networks and graph limits , volume 60 of Amer. Math. Soc. Colloq. Publ. American Mathematical Society, 2012. doi:10.1090/coll/060

    2012 doi

  11. [19]

    Weakly norming graphs are edge-t ransitive

    Alexander Sidorenko. Weakly norming graphs are edge-t ransitive. Combinatorica, 40(4):601– 604, 2020. doi:10.1007/s00493-020-4468-3

    2020 doi

  12. [20]

    Bipartite subgraphs and q uasi-randomness

    Jozef Skokan and Lubos Thoma. Bipartite subgraphs and q uasi-randomness. Graphs Combin., 20(2):255–262, 2004. doi:10.1007/s00373-004-0556-1

    2004 doi

  13. [21]

    An information theoretic approach to Sidorenko’s conjecture

    Bal´ azs Szegedy. An information theoretic approach to Sidorenko’s conjecture. Preprint avail- able at arXiv:1406.6738. 30

    arXiv

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