Writing study: HATSolver: Learning Gröbner Bases with Hierarchical Attention Transformers

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← ICLR 2026 oral writing corpus

Mohamed Malhou, Ludovic Perret, Kristin Lauter · Official ICLR 2026 oral record

Observed writing choices, not acceptance rules. This study examines the inspected manuscript version. It does not establish why the paper was accepted or that its presentation choices should be copied unchanged.

Figure, table and section ledger

Editorial contribution type: Architecture & representation design (primary).

Layout extraction, not a required paper template. Counts are distinct caption labels; a multi-panel figure counts once. Section spans are estimated from heading positions, including floats and whitespace. Method and limitations.

Canonical PDF

21 physical pages

Before references / appendix

≈ 10.16 page-equivalents

Main figures / tables

2 / 3

Attached appendix figures / tables

4 / 9

Attached appendix span

≈ 8.43 page-equivalents

Before references / appendix: ≈ 48.38%References / intervening material: ≈ 11.48%Attached appendix: ≈ 40.14%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Section-by-section lengths and source pages

The title/abstract/front matter precede the first numbered heading. A span ends at the next detected top-level heading, not at the last sentence; end-page whitespace is included. References and statements are listed separately. These are not measurements of text-only or artwork area.

Top-level heading	Starts on PDF page	Estimated span	Share of entire PDF
1 INTRODUCTION	1	≈ 1.29 pp.	6.14%
2 PRELIMINARIES	2	≈ 1.51 pp.	7.19%
3 METHOD	4	≈ 2.21 pp.	10.52%
4 EXPERIMENTS	6	≈ 3.03 pp.	14.43%
5 RELATED WORK	9	≈ 0.93 pp.	4.43%
6 CONCLUSION	10	≈ 0.28 pp.	1.33%
7 REPRODUCIBILITY STATEMENT	10	≈ 0.38 pp.	1.81%
REFERENCES	11	≈ 2.42 pp.	11.52%
A MATHEMATICAL FOUNDATIONS	13	≈ 0.53 pp.	2.52%
B CURRICULUM LEARNING SCHEDULERS	14	≈ 0.67 pp.	3.19%
C COMPLEMENTARY EXPERIMENTS	14	≈ 4.06 pp.	19.33%
D DATASETS	18	≈ 1.92 pp.	9.14%
E CLASSICAL ALGORITHM RUNTIME	20	≈ 0.57 pp.	2.71%
F TRAINING HYPERPARAMETERS	21	≈ 0.67 pp.	3.19%

All detected figure and table locations

Main text

• Figure 1 · p. 4
• Figure 2 · p. 7
• Table 1 · p. 8
• Table 2 · p. 8
• Table 3 · p. 9

Attached appendix

• Table 4 · p. 15
• Figure 3 · p. 15
• Figure 4 · p. 16
• Figure 5 · p. 17
• Figure 6 · p. 18
• Table 5 · p. 18
• Table 6 · p. 19
• Table 7 · p. 19
• Table 8 · p. 20
• Table 9 · p. 20
• Table 10 · p. 20
• Table 11 · p. 21
• Table 12 · p. 21

Argument and organization

1. Algebraic setting and encoding

   Defines the restricted problem before presenting a learnable replacement.

2. Hierarchical attention and cost

   Connects the input tree to bottom-up/top-down operations and complexity.

3. Scaling and ablations

   Separates curriculum, hierarchy depth, field and generator effects.

Problem, claim and contribution

The introduction identifies sequence-length explosion as the practical limitation of prior learned Gröbner-basis solvers. The shape-position restriction and finite-field setting then constrain the ambitious solver framing before architecture is introduced.

§§1–2.4, pp.1–4 · Main paper

Argument and section order

The cost derivation retains projection and attention terms and identifies assumptions behind the regular-tree simplification. The experiment discussion then admits padding overhead, connecting an idealized complexity argument to an implementation cost.

§3.3, pp.5–6; §4.1, pp.6–7 · Main paper

Figures, captions and visual explanation

Figure1 maps literal coefficient/exponent tokens to terms, equations and a system. This small worked example is more instructional than an abstract attention box, and supplies the hierarchy used by the following equations.

Figure1 and §§3.1–3.2, pp.4–5 · Main paper

Tables, comparisons and evidence

Table1 distinguishes exact match, support and token accuracy, while classical solvers fail by a two-hour cutoff. Runtime excludes neural training and classical averages concern completed runs; these are not interchangeable notions of correctness or end-to-end efficiency.

Table1 and §4.2, pp.7–8 · Main paper

Appendix and reproducibility

AppendixD exposes the backward-generator distribution and the 400-term training cap; at full density it retains less than ten percent of generated systems. AppendixE independently rechecks classical timings instead of inheriting a favorable prior baseline.

Tables7–11, pp.19–21 · Main paper

Clarity and writing conventions

Several final-version details need reconciliation: the curriculum uses sigma2 and pace1/2 in §4.2 but sigma4 and ramp1 in Table12; Msolve runtime/success prose differs from Table1. The conclusion also calls non-prime fields future work despite reporting them.

§§4.2 and6, pp.7–10; Table12, p.21 · Main paper

What to reuse—and what not to infer

Transferable lesson: Introduce structured input with one worked diagram, then pair complexity with measured padding and resource costs. Carry the data-generation restrictions into every solver-performance claim.

Boundary: The method does not solve arbitrary polynomial systems or demonstrate unseen-field transfer. Separately trained fields are not cross-field generalization, and the no-curriculum ablation also changes data exposure, so it does not isolate curriculum alone.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-09-19

Reading status

Full text + selected visuals inspected

Main text

pp.1–11, §§1–7; references pp.11–13

Appendix / supplement

pp.13–21, A–F: mathematical definitions, curriculum, additional experiments, datasets, runtime and training

Visually inspected PDF pages

4, 8

• official ICLR 2026 proceedings PDF · 21 pages