Geometric Generality of Transformer-Based Gr\"obner Basis Computation

Tristan Vaccon; Yota Maeda; Yuta Kambe

REVIEW 5 major objections 4 minor 38 references

Geometric Generality of Transformer-Based Gr\"obner Basis Computation

T0 review · 5 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Under an unproved irreducibility heuristic, the outputs of the proposed generator-sampling algorithm are Zariski dense in the space of generator sets of a fixed ideal, supplying a conditional geometric notion of dataset generality for transformer training.

desk verdict A solid conditional density theorem for a generalized generator construction, but the abstract overclaims: the result does not cover the restricted output set of the KIK+24 dataset, and the key heuristic is unproven. read the letter →

arxiv 2504.12465 v1 pith:KWPJGK4J submitted 2025-04-16 cs.LG cs.SCmath.AGstat.ML

Yuta Kambe , Yota Maeda , Tristan Vaccon This is my paper · ORCID

classification cs.LGcs.SCmath.AGstat.ML

keywords datasetsobnerbasiscomputationtrainingtransformersaddressalgorithm

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

To train a transformer to compute Gröbner bases, you need many pairs: a list of polynomials as input, and the Gröbner basis of the ideal they generate. Since computing Gröbner bases is expensive, a common trick is to start from a known basis and randomly build polynomial systems that generate the same ideal. Whether the random systems cover the full space of possible inputs has remained heuristic.

This paper formalizes 'covers the full space' using Zariski density, a topology from algebraic geometry where a set is dense if every nonzero polynomial condition that vanishes on the set vanishes everywhere. The authors consider the algorithm that applies a product of random elementary row operations to a fixed basis. They prove that, provided a certain algebraic set X≤D is irreducible (their Heuristic 2.2), and the field is Hilbertian, the possible outputs are Zariski dense. The proof uses classical theorems: Quillen-Suslin on free modules, Suslin's stability theorem, and Hilbert's irreducibility theorem.

The gap between this conditional theorem and the abstract's learning claim is real. Zariski density says the support of the generated examples is generic; it does not say a finite sample trains a transformer, and the theorem's range m≥2n lies mostly outside the m≤n+2 setting of the earlier experiments. The abstract says the result covers the previously proposed algorithm, but the theorem is proved for a more general class; density of a superset does not imply density of the original subset.

Extended reading notes

Core claim

Theorem 2.3 (Corollary 4.12): for a Hilbertian field K, m≥2n≥3, and under Heuristic 2.2 for D and G, the set F0 ∩ F~≤D is dense in F~≤D for the relative Zariski topology; consequently, if the heuristic holds for all sufficiently large D, F0 is Zariski dense in F = {F ∈ R^m | ⟨F⟩ = ⟨G⟩}. If the paper is correct, the possible outputs of the generalized generator algorithm form a generic set in the coefficient space, not a special subvariety.

Load-bearing premise

Heuristic 2.2 (Section 2.2): for given D and G, the algebraic set X≤D = {(B,A) ∈ R^{n×m}_{≤D} × R^{m×n}_{≤D} | (BA−En)G=0, B1A1∈R^{n×n}_{≤D}} is irreducible. The proof of Theorem 2.3 needs X≤D irreducible to apply Lemma 4.10 and Theorem 4.11; if the heuristic fails for the relevant D and G, the density conclusion is not established. The paper gives no proof or evidence for this irreducibility. Independently, the application to the earlier KIK+24 algorithm assumes its specific Bruhat-like output set is dense, a statement not proven in the paper.

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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

Worth a look if you care about ML-for-symbolic-math theory, but read it with the abstract flagged. The paper's real contribution is a conditional Zariski-density theorem for a broad class of generator tuples: for m≥2n≥3 and a Hilbertian field, and assuming Heuristic 2.2, the outputs of their Algorithm 2 are dense in the set of all generator tuples of a given ideal. The proof is built from standard but non-obvious ingredients—Quillen-Suslin, Suslin's stability theorem, and Hilbert's irreducibility—and the treatment of the determinant as an irreducible polynomial in Lemma 4.8 is credible. That is genuinely new.

The soft spots are in the packaging. The abstract and Remark 2.5 claim the density result applies to the dataset generation algorithm of KIK+24. It doesn't, as far as the paper shows. Theorem 2.3 concerns F0 = {U[En;0]G | U ∈ E(m)}, the output set of the new Algorithm 2. The KIK+24 construction restricts to a Bruhat-like family U1 S (U2;0) with triangular unipotent blocks, which is a proper subset of that F0. Zariski density of a superset says nothing about the subset, and the paper contains no proof that the restricted family is itself dense. The parameter mismatch makes the gap concrete: the theorem needs m≥2n, while the experiments in KIK+24 use m≤n+2, so the overlap is only n=2, m=4. Remark 2.6 admits this but still leans on the theorem for support.

Second, the result is conditional on Heuristic 2.2, an irreducibility statement about X≤D that is substantial and unproven. The paper gives no evidence. That is fine for a theorem if the statement is honest, but the abstract omits the condition entirely. The introduction does say 'assuming a heuristic,' so the authors are not hiding it from a careful reader.

Net: the conditional theorem is a meaningful step, and the algebraic machinery is used seriously. But the advertised geometric justification for the actual KIK+24 training data is not established. This deserves a real referee, not a desk reject, because the core proof is plausibly correct and the overclaim is fixable. I would ask the authors to either prove density for the restricted Bruhat-like family or rewrite the claims to match what Theorem 2.3 actually shows. The heuristic also needs at least a discussion of plausibility, ideally a proof for relevant G.

Referee Report

5 major / 4 minor

Summary. This paper studies the geometric generality of training data for Transformer-based Gröbner basis computation. It formulates Problem 2.1: an algorithm that, for a fixed tuple G of polynomials, randomly outputs generators F of the ideal ⟨G⟩, such that the set F0 of all possible outputs is Zariski dense in the set F of all generator tuples of ⟨G⟩. The authors propose Algorithm 2, which outputs F = A G where A = U (En;0)^T and U is a finite product of elementary matrices over the polynomial ring R. They prove (Theorem 2.3/Corollary 4.12) that, over a Hilbertian field, for m ≥ 2n ≥ 3 and under Heuristic 2.2 (irreducibility of X≤D), the set F0 ∩ ilde F≤D is dense in ilde F≤D; Corollary 2.4 upgrades this to density of F0 in F when the heuristic holds for all sufficiently large D. The proof uses Quillen–Suslin, Suslin's stability theorem, and Hilbert's irreducibility theorem. The introduction and abstract further claim that this justifies the earlier dataset generation method of [KIK+24].

Significance. If the conditional theorem is correct, it provides a genuine geometric statement about a class of generator-construction algorithms: the possible outputs are Zariski dense rather than confined to a special subvariety. The use of Quillen–Suslin to replace left-regular matrices by an elementary-matrix action is elegant, and the section construction in Corollary 4.12 is a good idea. The paper is also honest in labelling Heuristic 2.2 as an assumption, and the proof is mostly self-contained. However, the significance as stated in the abstract is not achieved: the theorem concerns the new generalized Algorithm 2, not the specific Bruhat-like construction of [KIK+24], and it applies in a parameter regime (m≥2n) that is disjoint from the experiments except for n=2,m=4. The unproven Heuristic 2.2 and the flat-locus proof gap further limit the current version.

major comments (5)

[Abstract, §2.2, §3.1] The advertised claim that the dataset generation algorithm of [KIK+24] is shown to be sufficiently general is not supported by the theorems. Theorem 2.3 and Corollary 4.12 prove that F0 = {U(En;0)^T G | U ∈ E(m)} is dense in F, and Proposition 4.2 identifies this set with all outputs F=AG for left-regular A over R. The construction of [KIK+24] recalled in §3.1 uses the restricted Bruhat-like family A = U1 S (U2;0)^T. Since F0 is a superset of the KIK output set, density of F0 says nothing about density of the restricted family. No statement or proof in the paper establishes that the Bruhat-like family is itself Zariski dense in F, and the missing logical bridge is not repaired by Proposition 4.2, which only characterizes the larger class.
[Remark 2.6 / Theorem 2.3] There is a parameter-regime mismatch: the main theorem assumes m ≥ 2n ≥ 3, whereas Remark 2.6 states that the experiments in [KIK+24] impose m ≤ n+2. For n ≥ 3 these conditions are incompatible; the only overlapping case is (m,n) = (4,2). Thus even if Heuristic 2.2 were fully justified, the theorem would not cover the experimental regime used in the prior work. The authors should prove a version for m < 2n or explicitly restrict their claims to the new setting.
[Heuristic 2.2] The core density theorem is conditional on Heuristic 2.2, the irreducibility of X≤D, for which the paper offers no proof, no evidence, and no discussion of regimes where it might fail. Since this assumption is load-bearing for Lemma 4.10 and Theorem 4.11, and hence for Theorem 2.3, the paper's central conclusion is only an implication from an unverified algebraic-geometric statement. At minimum, the authors should prove or test irreducibility in nontrivial cases such as n=2, m=4, or give a structural argument based on the equations (BA−En)G=0.
[Corollary 4.12 / Theorem 4.11] The proof that the flat locus Y is nonempty contains a gap. The sentence 'Let L be the field of fractions ... Then Spec L is isomorphic to a non-empty open subscheme V in R^{n×n}_{≤D}' is not correct: Spec L is the generic point and is not an open subscheme of a positive-dimensional affine space. Consequently the claim Y ≠ ∅ is not established as written. The gap appears repairable by invoking generic flatness for finite-type morphisms to a reduced Noetherian scheme, which gives a nonempty open V over which p is flat, but the current text does not supply that argument.
[Problem 2.1 / Abstract] Problem 2.1 and the abstract formulate the result in terms of datasets being dense, but a finite training dataset is never Zariski dense in an infinite variety. What the proof actually establishes is that the set of all possible outputs of Algorithm 2 is dense in the set of all generators of ⟨G⟩. This distinction matters for the claimed learning guarantee: density of the generator's support is only a necessary condition for a finite sample to be representative, not a sufficient one. The paper should state this limitation explicitly.

minor comments (4)

[§4.2] In the displayed definition of ilde F≤D before Lemma 4.10, the notation X_{≤D}^{m×n} is inconsistent with the definition of X_{≤D} in Heuristic 2.2; this appears to be a typo.
[Lemma 4.8] In the proof, the displayed factorization uses the symbol u3 twice in the second factor; this is presumably a typo and should involve two distinct coefficients.
[§3.1] There is a typo: 'basss' should be 'bases'.
[§2.1] The notation F = {F ∈ R^m | ⟨F⟩ = ⟨G⟩} treats G as fixed, while Problem 2.1 presents G as an input; please make the dependence on G explicit in the notation throughout.

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

All parameters (m, n, r, D, G) are inputs, not fitted values. The proof relies on standard theorems from algebraic geometry and commutative algebra, plus one unproved ad hoc heuristic, Heuristic 2.2, on the irreducibility of X≤D. No new entities are postulated.

assumptions (6)

standard math Quillen-Suslin theorem (Serre's conjecture): finitely generated projective modules over K[x1,...,xr] are free.
Invoked in Proposition 4.2 to identify the kernel of phi_B as free and to obtain the equivalence of split exact sequences.
standard math Suslin's stability theorem: for m>=3, E(m) = <GL(K^m), SL(R^m)>.
Used in Proposition 4.2 to express the isomorphisms U and V as products of elementary matrices.
domain assumption Hilbert's irreducibility theorem holds for Hilbertian fields K.
Used in Lemma 4.9 to show matrices whose determinants are non-zero irreducible polynomials form a Zariski-dense set; Theorem 2.3 assumes K is Hilbertian, e.g., number fields.
ad hoc to paper Heuristic 2.2: X≤D is irreducible for the given D and G.
This is the unproved assumption on which the density theorem is conditional; the flat-locus argument needs an irreducible domain.
standard math Flat morphisms of finite type between Noetherian schemes are open (Hartshorne III.9.1).
Used in Lemma 4.10 to show that the preimage of a dense set under a flat morphism is dense.
domain assumption Parameter constraints m>=2n>=3 and R=K[x1,...,xr] with K Hilbertian.
The theorem and proof require these assumptions; they exclude most experimental settings from KIK+24 where m<=n+2.

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Pith. "Pith review of Geometric Generality of Transformer-Based Gr\"obner Basis Computation." pith.science (2026). https://pith.science/paper/KWPJGK4J

@misc{pith2026250412465,
  author       = {Pith},
  title        = {Pith review of: Geometric Generality of Transformer-Based Gr\"obner Basis Computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KWPJGK4J}},
  note         = {Machine review of arXiv:2504.12465}
}

read the original abstract

The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical problems critically depends on high-quality, domain-specific datasets. In this paper, we address the computation of Gr\"obner basis using Transformers. While a dataset generation method tailored to Transformer-based Gr\"obner basis computation has previously been proposed, it lacked theoretical guarantees regarding the generality or quality of the generated datasets. In this work, we prove that datasets generated by the previously proposed algorithm are sufficiently general, enabling one to ensure that Transformers can learn a sufficiently diverse range of Gr\"obner bases. Moreover, we propose an extended and generalized algorithm to systematically construct datasets of ideal generators, further enhancing the training effectiveness of Transformer. Our results provide a rigorous geometric foundation for Transformers to address a mathematical problem, which is an answer to Lample and Charton's idea of training on diverse or representative inputs.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Roberto Avanzi, Joppe Bos, L \'e o Ducas, Eike Kiltz, Tancr \`e de Lepoint, Vadim Lyubashevsky, John M Schanck, Peter Schwabe, Gregor Seiler, and Damien Stehl \'e , CRYSTALS-Kyber (version 3.02) , Submission to round 3 of the NIST post-quantum project, Available at https://pq-crystals.org/

work page

[2] [2]

Alberto Alfarano, Fran c ois Charton, and Amaury Hayat, Global lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers, Advances in Neural Information Processing Systems 37 (2024), 93643--93670

work page 2024

[3] [3]

ACC (IEEE Cat

Hirokazu Anai and Shinji Hara, Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination , Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No. 00CH36334), vol. 2, IEEE, 2000, pp. 1312--1316

work page 2000

[4] [4]

B \^o cher, Introduction to Higher Algebra , Macmillan, 1907

M. B \^o cher, Introduction to Higher Algebra , Macmillan, 1907

work page 1907

[5] [5]

Proceedings, Springer, 2006, pp

Johannes Buchmann, Andrei Pyshkin, and Ralf-Philipp Weinmann, Block ciphers sensitive to G r \"o bner Basis Attacks , Topics in Cryptology--CT-RSA 2006: The Cryptographers’ Track at the RSA Conference 2006, San Jose, CA, USA, February 13-17, 2005. Proceedings, Springer, 2006, pp. 313--331

work page

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Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, SIGSAM Bull

Bruno. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, SIGSAM Bull. 10 (1976), no. 3, 19–29

work page 1976

[7] [7]

Leandro Caniglia, Guillermo Corti \ n as, Silvia Dan \'o n, Joos Heintz, Teresa Krick, and Pablo Solern \'o , Algorithmic aspects of suslin's proof of serre's conjecture, computational complexity 3 (1993), 31--55

work page 1993

[8] [8]

David Cox, John Little, and Donal O'Shea, Ideals, Varieties, and Algorithms , second ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997, An introduction to computational algebraic geometry and commutative algebra. 1417938

work page 1997