Galois groups of polynomials and neurosymbolic networks

Elira Shaska; Tony Shaska

REVIEW 5 major objections 6 minor 22 references

Galois groups of polynomials and neurosymbolic networks

T0 review · 5 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a neurosymbolic network—a feedforward classifier wrapped in symbolic layers for mod-$p$ signatures, discriminants, and real-root counts—classifies Galois groups of low-degree polynomials more efficiently than a plain…

desk verdict The abstract's efficiency claim is never tested: Section 5 describes a GaloisNetwork but reports no accuracy, baselines, or training metrics, so the paper's central novelty is unsubstantiated. read the letter →

arxiv 2501.12978 v1 pith:DJY7OAJQ submitted 2025-01-22 cs.LG cs.AImath.HO

Elira Shaska , Tony Shaska This is my paper · ORCID

classification cs.LGcs.AImath.HO MSC 11R3268T0712Y0511R09

keywords GaloistheoryNeurosymbolicAInetworksgroupclassificationpolynomialdatabasesboundedheightinvariantsolvabilitybyradicals

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 sets out to show that Galois groups of polynomials can be classified by a neurosymbolic network: a standard feedforward classifier surrounded by classical symbolic layers for reduction modulo $p$, discriminant square-testing, and real-root counting. The authors build exact databases of irreducible cubics, quartics, and quintics of bounded height whose Galois groups are known, then use them to train and test the hybrid model. The databases yield a concrete number-theoretic discovery: cyclic Galois groups are extremely rare at small height—for example, exactly 20 irreducible quintics of height at most 10 have Galois group $C_5$, in three $\mathbb{Q}$-isomorphism classes. If the central claim is correct, this architecture is a template for pushing Galois-group computation to degrees where pure symbolic methods slow down and pure learning cannot see rare groups.

What carries the argument

The central object is the GaloisNetwork, a neurosymbolic classifier that combines a feedforward network (input layer, three hidden layers of 64 ReLU units, softmax output) with symbolic layers that encode classical Galois theory. The signature layer reduces the polynomial modulo $p = 2,3,5,7$, records the factorization degree pattern, and compares it against the precomputed list of transitive-subgroup signatures, fixing the group when exactly one candidate remains. The discriminant layer checks whether $\Delta_f$ is a square in $\mathbb{Q}$, separating subgroups of $A_d$ from those that are not. The real-root layer counts real roots by sign variations and applies the theorem that a prime-degree polynomial with few non-real roots must have Galois group $A_p$ or $S_p$. The training databases are built by enumerating projective points of bounded height, keeping irreducible forms, and labelling each by its invariants and Galois group; for quintics the invariants $J_4, J_8, J_{12}, J_{18}$ and the resolvent quantities $d_1,\dots,d_6$ carry the solvability criterion.

What would settle it

Recompute the bounded-height enumeration with an independent implementation: for degree 5, height at most 10, list all irreducible integer quintics with projective height $\leq 10$, compute their Galois groups by a resolvent-based method, and compare the counts. The paper predicts exactly 20 polynomials with Galois group $C_5$ in three $\mathbb{Q}$-isomorphism classes, 480 with $F_5$, 900 with $D_5$, and 1,146 with $A_5$; for quartics it predicts 5,676 irreducible polynomials of height $\leq 10$ not having Galois group $S_4$. Any mismatch in these exact totals would show the database, and therefore the training set, is unreliable.

Watch

Extended reading notes

Core claim

The paper's central claim is that a hybrid network whose neural predictions are filtered through symbolic checks outperforms a plain neural network for classifying Galois groups of irreducible polynomials over $\mathbb{Q}$ in small degrees. The symbolic checks are the factorization signature of $f(x) \bmod p$ for $p = 2,3,5,7$, the question of whether the discriminant $\Delta_f$ is a square, and the number of real roots obtained by sign-variation counting; the paper reports that these layers, combined with the invariants of binary forms, resolve the Galois group directly in many cases and leave only genuinely ambiguous polynomials to the neural head. The supporting databases are exact enumerations of projective points of bounded height, filtered to irreducible polynomials and labelled by Galois group. On the basis of those databases the paper states exact counts, including 5,676 irreducible quartics of height $\leq 10$ with Galois group not isomorphic to $S_4$ (5162 $D_4$, 184 $A_4$, 222 $V_4$, 108 $C_4$) and, for quintics of height $\leq 10$, exactly 20 with Galois group $C_5$, 480 with $F_5$, 900 with $D_5$, and 1,146 with $A_5$, with the $C_5$ examples falling into three $\mathbb{Q}$-isomorphism classes.

Load-bearing premise

The whole edifice rests on the assumption that the computer enumeration of bounded-height irreducible polynomials is complete and that every Galois label attached to a polynomial is correct; a single wrong count or wrong label would corrupt the rarity discovery and every training example built from that data.

Editorial extensions

If this is right

When the mod-$p$ signature is unique to one transitive subgroup, the symbolic layer fixes the Galois group before learning is needed, so the neural component only has to handle the remaining ambiguous polynomials.
The exact rarity counts give a concrete check for distribution conjectures about bounded-height polynomials, such as the expected frequency of cyclic groups and recent bounded-invariant counts for quartic forms.
For degrees 6 and 7, where transitive subgroup lists are much larger and signatures are no longer unique, the same three-layer architecture is the paper's proposed route for keeping classification tractable.
Using absolute invariants to collapse $\mathbb{Q}$-equivalent polynomials shrinks the effective dataset and, according to the paper, makes the learning problem easier rather than harder.

Reading between the lines

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

A natural next experiment is a head-to-head accuracy-versus-training-size comparison with a plain network on identical splits; the efficiency claim in the paper is architectural and such a curve would make it quantitative.
If the enumeration is exact, the 20-quintic count implies that a purely statistical learner would need to see nearly the whole height-$\le 10$ population to collect even a handful of $C_5$ examples; the symbolic layers supply exactly the inductive bias that makes such rare classes learnable.
The same modular recipe—cheap symbolic necessary conditions plus a neural head—transfers naturally to other classification problems in algebraic geometry where the paper's moduli-height machinery already applies, such as automorphism groups or isogeny classes.
A direct scaling test would be to run the same architecture on sextics and septics with identical height bounds, since the paper identifies those degrees as the point where symbolic computation alone begins to slow down.

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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

Here's the bottom line: the paper's headline claim—that a neurosymbolic network classifies Galois groups more efficiently than a plain neural network—is not supported by anything in the text. Section 5 describes the architecture and says it was implemented and tested, but gives no numbers, no training curves, no baseline comparison, no wall-clock time. That is not a minor omission; it's the entire stated contribution.

What the paper does well is the classical background. The cubic and quartic resolvent criteria are correctly presented and properly attributed, and the quintic solvability discussion via the sextic resolvent is accurate as far as it goes. The idea of creating height-bounded databases of irreducible polynomials with their Galois groups is reasonable, and the reported counts (e.g., 20 quintics of height ≤ 10 with C5) would be genuinely useful if the enumeration were reproducible. The tables of transitive subgroups are also handy.

The soft spots are substantial. First, the enumeration is unverifiable: no code, no data snapshot, no commit hash. The description relies on Sage's rational_points(h) without spelling out the normalization from projective points to integer polynomials, and the tables contain duplicated entries (e.g., Table 5). Second, the proof of Lemma 8 is not valid as written. The displayed formula for Δ_g includes factors I_2, I_3, I_4, I_6 that could vanish even when Δ_f ≠ 0, so Δ_f ≠ 0 does not imply Δ_g ≠ 0. The lemma may be true classically, but this proof doesn't establish it. Third, the Appendix code for the signature layer is not a correct implementation of Dedekind's theorem: it never checks that the primes don't divide the discriminant, and it stores a set of factor degrees, discarding multiplicities that are required.

None of these issues are fatal to the underlying idea—a neurosymbolic classifier for Galois groups could be useful, especially for degrees ≥ 6. But the paper as written is a survey plus a proposal, not a demonstrated result. The audience that would get value is someone wanting a quick refresher on the classical criteria and a cautionary example of how an ML paper can overclaim. For a serious referee, it's not ready: the main claim is untested and the data is not checkable. My recommendation is to desk reject, with a note that adding real experiments, a reproducible data release, and a fixed proof of Lemma 8 would make a resubmission worth considering.

Referee Report

5 major / 6 minor

Summary. The paper proposes a neurosymbolic architecture for classifying Galois groups of irreducible integer polynomials of degrees 3–5, combining a feedforward ReLU network with symbolic layers based on real-root counts, reductions modulo primes, and discriminant checks. It also reports enumeration counts and distributions for height-bounded cubic, quartic, and quintic datasets, including a table of the twenty quintics of height at most 10 with Galois group C5. The theoretical sections review classical criteria for Galois groups of cubics, quartics, and quintics, and Section 5 sketches an implementation called GaloisNetwork. The central efficiency claim—that the neurosymbolic network is more efficient than plain neural networks—is not supported by any reported experiments.

Significance. If the claimed database counts and invariant formulas were fully established, they would provide useful concrete test data for questions around Malle's conjecture and for comparing naive and moduli heights. The explicit enumeration of height-bounded quintics with Galois group C5 in Table 4 is a concrete, potentially checkable contribution. However, the paper's main advertised novelty, the neurosymbolic efficiency gain, has no empirical support, and a load-bearing proof in the solvability criterion is incomplete. The paper does not ship reproducible code or a data snapshot, so the exact counts cannot be independently verified. Credit is due for the explicit invariant formulas and the attempt to combine Dedekind's theorem with learned features, but as written these are classical ingredients assembled into an untested sketch rather than a demonstrated result.

major comments (5)

[Section 1 / Section 5] The abstract's central claim that the neurosymbolic network is 'more efficient than usual neural networks' is not supported by any measurement in the paper. Section 5, 'Implementation and efficiency', describes a 64-64-64 ReLU MLP with a softmax output and a post-processing rule, but it reports no accuracy, loss curves, confusion matrices, wall-clock time, parameter counts, or any baseline comparison to a plain neural network. Section 1 is even more explicit, calling the system 'not a fully developed product yet' and saying there 'might not be a compelling reason to use AI models' for degrees 3, 4, and 5; this directly contradicts the definitive efficiency claim in the abstract.
[Section 2.3, Lemma 8] The proof of Lemma 8 infers Delta_g != 0 from Delta_f != 0 via the displayed product formula Delta_g = ((alpha1-alpha2)(alpha3-alpha4)(alpha4-alpha5)(alpha3-alpha5))^4 * Delta_f * I_2^2 * I_3 * I_4^2 * I_6^2, but the nonvanishing of the invariants I_2, I_3, I_4, and I_6 is never established. If any of these invariants vanishes on a polynomial with distinct roots, the product formula does not imply Delta_g != 0, and the lemma is not proven. The formula is introduced as a result of 'computational algebra' without a derivation, an implementation, or a reference, so the reader cannot verify it.
[Section 4, Lemmas 11-13] The enumeration counts and the training data depend on the Sage call 'PP.rational_points(h)' followed by an irreducibility filter, but the manuscript provides no code, data snapshot, or commit hash, and the exact normalization, duplicate-removal, and projective-representative conventions are not fully specified. Consequently, claims such as 1,178,856 irreducible cubics, 5,676 non-S4 quartics, and exactly 20 quintics with Galois group C5 cannot be checked. An error in enumeration, scaling, or class labeling would invalidate both the distribution discoveries and every trained example in Section 5.
[Section 5, Listing 1] The signature layer as implemented in Listing 1 does not check that the chosen primes satisfy p does not divide Delta_f, which is required for the validity of Dedekind's theorem. It also stores only factor degrees as a set, discarding multiplicities and treating repeated factors of the same degree as a single entry. The resulting 'signature' can therefore disagree with the group signatures defined in Section 3.2, so the symbolic layer's correctness is not established.
[Section 4.2 / Section 5] The manuscript states that Galois labels are computed by an external computer algebra system and that the features are classical invariants (discriminant, real-root count, mod-p factorization). This means the model itself does not discover or define the target; the claimed advantage over a plain network must be demonstrated by a controlled comparison. Since no such comparison appears, the paper's title and abstract overstate what is currently a proposal plus a database, not a validated method.

minor comments (6)

[Section 2.3, Lemma 7] The statement of Lemma 7 contains the Albanian words 'dhe' and 'për' in an otherwise English mathematical text; these should be replaced with 'and' and 'for'.
[Sections 4.2 and 4.3] The phrase 'the ration of the weighted height over the naive height' should read 'the ratio of the weighted height to the naive height' in both occurrences.
[Appendix A.1, Table 5] Table 5 has duplicate row numbers and several rows without an explicit discriminant value, so the list of forty cubics with Galois group C3 is not usable as a clean data table.
[Section 3.1, Table 1] Table 1 lists the number of transitive subgroups of S_n for n up to 45 but omits n = 32, which is needed for the claimed range and for checking the completeness of the enumeration.
[Section 4, definitions] The weighted moduli height, the set P_h^n, and the invariants used in the database are defined by reference to [16] and [17] rather than in the paper, making Section 4 not self-contained and the exact height convention ambiguous.
[Section 4.2, invariants] In Section 4.2, J6 is first defined as Delta(f,x) and then stated to satisfy J6 = (4*J2^3 - J3^2)/27; the relationship between these two definitions should be stated explicitly to avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Galois labels come from external computer algebra, features are classical invariants, and the reported counts are enumerations rather than fitted predictions.

full rationale

The paper's derivation chain is not circular. The Galois group labels in the Section 4 databases are computed externally via Sage's rational_points and GalQ(f), not by the proposed model; the features used (discriminant, invariants, mod-p signature, real-root count) are classical invariants independent of the labels. The quintic solvability criterion (Lemma 10) is quoted from the classical Berwick formulas and is not refitted here. The symbolic layer uses Dedekind's theorem to compare factor-degree signatures against group signatures, which is an independent mathematical constraint rather than a definition of the target. The abstract's efficiency claim is indeed unsupported because Section 5 reports no accuracy, baseline, or wall-clock comparison, but that is a missing-evidence/correctness problem, not circularity. Self-citations [15,16,17] supply height definitions, invariant theory, and implementation details; these are parameter-free tools that do not assume the target result, so under the stated rules they do not raise the circularity score beyond a minor note.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper leans on a large body of classical invariant theory and computational algebra. Its own contribution rests on hand-chosen enumeration bounds, hand-picked symbolic-layer parameters, an unproven algebraic identity in Lemma 8, and an unverified database enumeration.

free parameters (4)

Height bound h for databases = h = 10 (quintics), h = 20 (cubics, quartics)
All distribution counts depend on this hand-chosen cutoff; changing it changes the reported frequencies (Sections 4.1-4.3).
Primes for signature layer = p = 2, 3, 5, 7
The signature layer's factorization output and thus the candidate-group filtering depend on this hand-chosen prime list (Appendix A.3, Listing 1).
Network hyperparameters = 3 hidden layers, 64 neurons, 100 epochs, Adam
The network architecture is chosen by hand and not optimized or compared; no accuracy is reported (Section 5).
Real-root evaluation endpoints = +/- 10^10
The real-root count uses finite endpoints to approximate infinity; only roots inside the interval are counted correctly (Appendix A.3, Listing 2).

assumptions (5)

standard math Standard results in Galois theory: transitivity of the Galois group for irreducible polynomials, discriminant square criterion, resolvent theory, and the classification of transitive subgroups of S5 (C5, D5, F5, A5, S5).
Invoked throughout Sections 2-3 (Lemmas 1-6, Theorem 3.1).
standard math Berwick's formulas expressing d_r in terms of the invariants J4, J8, J12 (equations in Section 2.3.1).
Used in Lemma 10 to state the sextic-resolvent solvability criterion; the paper cites [8] but does not derive the formulas.
ad hoc to paper The discriminant formula Delta_g = ((alpha1-alpha2)(alpha3-alpha4)(alpha4-alpha5)(alpha3-alpha5))^4 * Delta_f * I_2^2 I_3 I_4^2 I_6^2 and the inferred nonvanishing of the I-factors.
Lemma 8's proof relies on this computational-algebra identity and on the unsupported claim that Delta_f != 0 implies Delta_g != 0, since the extra I-factors could vanish.
domain assumption Completeness of the Sage enumeration 'PP.rational_points(h)' and correctness of all Galois labels computed by the computer algebra system.
Section 4 builds P_h^n and derives all distribution counts from it; no code or data artifact is shipped to verify the enumeration.
domain assumption The heuristic that 'there is no universal method in Galois theory that works for every degree' justifies per-degree networks.
Section 1 uses this to motivate the neurosymbolic design; it is a heuristic, not a theorem.

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Cite this review

Pith. "Pith review of Galois groups of polynomials and neurosymbolic networks." pith.science (2026). https://pith.science/paper/DJY7OAJQ

@misc{pith2026250112978,
  author       = {Pith},
  title        = {Pith review of: Galois groups of polynomials and neurosymbolic networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJY7OAJQ}},
  note         = {Machine review of arXiv:2501.12978}
}

read the original abstract

This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline the process of determining solvability by radicals and explore broader applications within Galois theory. This summary encapsulates the background, methodology, potential applications, and challenges of using data science in Galois theory. More specifically, we design a neurosymbolic network to classify Galois groups and show how this is more efficient than usual neural networks. We discover some very interesting distribution of polynomials for groups not isomorphic to the symmetric groups and alternating groups.

Figures

Figures reproduced from arXiv: 2501.12978 by the authors.

**Figure 1.** Lattice of transitive subgroups of 𝑆5 The signature of 𝐺 is the set of such types of permutations. Notice that for quintics not two subgroups have the same signature. Unfortunately this is not the case for higher degree, so the signature does not always uniquely determines the group. 3.3 Polynomials with non-real roots Let 𝑓 (𝑥) ∈ Q[𝑥] be an irreducible polynomial of degree 𝑛 > 5. Denote by 𝑟 the number of non-real … view at source ↗

**Figure 2.** Distribution of cubics with Galois group [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Occurrences for cubics versus the invariants [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 7.** Figure 7: The ratio of weighted height with naive height [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 5.** Figure 5: The ratio of weighted height with naive height [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Distribution of quintics with Gal (𝑓 ) . 𝑆5. Next is presented the Python implementation of the symbolic layers for irreducible quintics: 1 from sympy import symbols , Poly , factor_list 2 def sig_layer ( p ) : 3 x = symbols ('x') 4 f = sum ( a * x ** i for i , a in en…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

W. E. H. Berwick. 1928. On Soluble Sextic Equations. Proc. London Math. Soc. (2) 29, 1 (1928), 1–28. https://doi.org/10.1112/plms/s2-29.1.1

work page doi:10.1112/plms/s2-29.1.1 1928

[2] [2]

Manjul Bhargava and Arul Shankar. 2015. Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Ann. of Math. (2) 181, 1 (2015), 191–242. https://doi.org/10.4007/annals.2015.181.1.3

work page doi:10.4007/annals.2015.181.1.3 2015

[3] [3]

Bialostocki and T

A. Bialostocki and T. Shaska. 2005. Galois groups of prime degree polynomials with nonreal roots. In Computational aspects of algebraic curves . Lecture Notes Ser. Comput., Vol. 13. World Sci. Publ., Hackensack, NJ, 243–255. https://doi.org/ 10.1142/9789812701640_0015

work page doi:10.1142/9789812701640_0015 2005

[4] [4]

Elira Curri. 2022. On the stability of binary forms and their weighted heights. Albanian J. Math. 16, 1 (2022), 3–23

work page 2022

[5] [5]

Tereso del Río and Matthew England. 2024. Lessons on datasets and paradigms in machine learning for symbolic computation: a case study on CAD.Math. Comput. Sci. 18, 3 (2024), Paper No. 17, 27. https://doi.org/10.1007/s11786-024-00591-0

doi:10.1007/s11786-024-00591-0 2024

[6] [6]

Igor Dolgachev. 2003. Lectures on invariant theory . Lond. Math. Soc. Lect. Note Ser., Vol. 296. Cambridge: Cambridge University Press

work page 2003

[7] [7]

Hagedorn

Thomas R. Hagedorn. 2000. General formulas for solving solvable sextic equations. J. Algebra 233, 2 (2000), 704–757. https://doi.org/10.1006/jabr.2000.8428

work page arXiv 2000

[8] [8]

Bruce King

R. Bruce King. 1996. Beyond the quartic equation. Birkhäuser Boston, Inc., Boston, MA. viii+149 pages

work page 1996