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Simple generators of rational function fields

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Pith's one-line read This paper presents a randomized algorithm that, given any finite set of generators of a subfield of a rational function field over a characteristic-zero field, finds a simpler generating set for the same field, and proves it terminates wit

desk verdict Strong engineering and benchmarks, but the proof of Algorithm 6's support recovery is unsupported: a single specialization can lose non-leading monomials, undermining the main correctness claim. read the letter →

arxiv 2602.10878 v3 pith:7UFCCY7Y submitted 2026-02-11 cs.SC cs.MScs.SYeess.SYmath.ACmath.DS

Alexander Demin , Gleb Pogudin This is my paper · ORCID
classification cs.SCcs.MScs.SYeess.SYmath.ACmath.DS MSC 68W3013P1012F20
keywords rationalfunctionfieldssimplegeneratorsGröbnerbasesOMSidealssparseinterpolationstructuralidentifiabilityinvariantsrandomizedalgorithms
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

The paper presents an algorithm that, given generators of a subfield of the rational function field k(x) over a characteristic-zero field, returns a simpler generating set of the same field. The key idea is that only low-degree coefficients of the Gröbner basis of the OMS ideal are needed, and these can be reconstructed by sparse interpolation from evaluations over finite fields, avoiding the expensive full Gröbner computation over rational functions. The algorithm also searches for low-degree polynomial elements of the subfield and filters candidates with randomized membership tests. A sympathetic reader should care because simplified generators turn unreadable parameter sets in applications like structural identifiability, observability of discrete-time systems, and rational invariant theory into interpretable expressions. The method is implemented and tested on 53 real-world examples, with a median runtime of about 20 seconds.

What carries the argument

The central object is the OMS ideal of the subfield E: given generators g_i=p_i/q_i, it is the ideal in k(x)[y] generated by p_i(y)q_i(x)−q_i(y)p_i(x), saturated by the common denominator Q(y). A classical fact (Lemma 2.12 here) says that the coefficients of the reduced Gröbner basis of this ideal generate E. The paper's mechanism is to recover those coefficients without computing the full basis: specialize x to random points, compute Gröbner bases over finite fields, and reconstruct the coefficients as sparse rational functions. Lemma 3.3 guarantees that for generic specializations the specialized Gröbner basis equals the specialization of the true basis, which is what makes the interpolati

What would settle it

Take any benchmark (e.g., the SLIQR model) and run the released implementation, then independently verify the output h by a second exact method (e.g., computing the Gröbner basis of the OMS ideal over Q for the output). If the Step-7 check ever returns True while k(h)≠k(g), the paper's correctness claim fails for the implementation; Proposition 4.1 would still hold only for the hypothetical version with an exact final check.

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Extended reading notes

Core claim

The paper claims that every finitely generated subfield E of k(x) (with k of characteristic zero) admits a simpler generating set that can be found by a randomized algorithm that never computes the full Gröbner basis of the OMS ideal. The central formal result (Proposition 4.1) states that Algorithm 8 terminates with probability one and, with probability at least 1−ε, outputs h with k(h)=k(g). The algorithm works by interpolating only the low-degree coefficients of the reduced Gröbner basis of the OMS ideal from evaluations over finite fields, then augmenting the candidate pool with low-degree polynomial elements of E, and finally filtering candidates using randomized membership tests. The p

Load-bearing premise

The whole pipeline assumes that every sampled evaluation is generic enough that the reduced Gröbner basis of the specialized ideal equals the specialization of the true Gröbner basis (Lemma 3.3), and that the sparse-interpolation subroutines receive correct degree bounds and only encounter FAIL rather than silent wrong values; moreover, the shipped implementation's final check over Q uses a Monte-Carlo multi-modular method with no practical error bound, so the implemented pro

Editorial extensions

If this is right

  • For any finitely generated subfield of k(x) over a characteristic-zero field, a simplified generating set can be obtained without ever computing the full Gröbner basis of the OMS ideal; only low-degree coefficients are recovered via sparse interpolation, which is what makes large problems tractable.
  • On a suite of 53 real-world examples from structural identifiability, the algorithm's median runtime is about 20 seconds; in 26 of the 53 cases it reveals that the subfield is generated by polynomials alone, and in all but one case the output is an algebraically independent generating set.
  • The simplified generators are small enough to interpret: e.g., a 3533-function generating set for a pharmacokinetics model collapses to 7 parameters, and an epidemiological model's 40 generators collapse to 6 expressions that expose the model's symmetry and natural rate combinations.
  • Applying the same simplification to outputs of cross-section-based invariant computation (e.g., for permutation group actions, triangular group actions, and rotation moment invariants) yields polynomial or low-degree rational generators that match or improve on classical lists.
  • The algorithm is implemented and available as an open-source package; for the benchmarks, the previous OMS-based simplification approach did not finish on 8 examples, while Algorithm 8 solved all but one of those within the same time and memory limits.

Reading between the lines

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

  • The 'partial Gröbner' idea is likely transferable: any computation over rational function fields that only needs low-degree output coefficients (GCDs, resultant entries, linear system solutions) could use the same evaluation-interpolation with early stopping, potentially beating full Gröbner approaches on the same class of inputs.
  • The polynomial-generator search (Algorithm 7) might serve as a building block for other subfield problems—e.g., computing intersections of subfields, testing algebraic dependence, or finding a transcendence basis containing low-degree polynomials—none of which the paper explores.
  • The algorithm's output depends on the heuristic order for 'simple' and on the parameter δ; feeding the simplified set back into the algorithm (using it as the new input) is a natural self-improving loop that the paper does not test, and it could yield even smaller generators on hard examples.
  • The theoretical guarantee is contingent on the final exact check over Q; a variant that carries out the entire computation over Q (or over several primes with a proven error bound) would be needed to make the 1−ε guarantee hold in practice.
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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

2 major / 5 minor

Summary. The paper presents Algorithm 8, which takes a finite generating set g of a subfield E of k(x), char k=0, and returns a 'simpler' generating set. The method uses OMS ideals, computes only low-degree coefficients of their reduced Gröbner bases by evaluation/interpolation (Algorithm 6), searches for low-degree polynomial elements (Algorithm 7), filters candidates by randomized membership tests (Algorithm 4), and uses a final randomized check to certify equality of fields. The paper proves termination with probability one and output correctness with probability at least 1−ε (Prop. 4.1), reports an open-source Julia implementation, and presents extensive benchmarks (53 problems) and case studies in identifiability, discrete-time systems, and invariant theory.

Significance. This is a well-motivated and potentially significant contribution. The high-level architecture is sound and the empirical results are striking: many input generating sets with dozens or hundreds of large rational functions are reduced to a handful of low-degree generators. The paper includes explicit randomized correctness bounds for its main subroutines, an open-source implementation with reproducibility data, and honest disclosure that the final check in the implementation has no proven probability bound (Remark 5.1). The applications to structural identifiability are convincing. However, the support-recovery step in Algorithm 6 is not covered by Lemma 3.3 as stated, so Proposition 3.4's probability bound is currently false; this is a load-bearing gap in the termination proof of Proposition 4.1. The issue is likely repairable, but the paper as written does not establish its sub-routine guarantee.

major comments (2)
  1. [§3.2, Algorithm 6 / Prop. 3.4] Algorithm 6 Step 2 recovers the support of the reduced Gröbner basis over k(x) from one specialization G_σ, citing Lemma 3.3. That lemma only guarantees equality of the specialized basis as a set of polynomials, not that non-leading coefficients are nonzero at σ. If c(σ)=0 for a non-leading coefficient, its monomial is absent from G_σ and is never interpolated. Example: E=Q(x1+x2,x1x2) has reduced GB {y1+y2−(x1+x2), y2^2−(x1+x2)y2+x1x2}; at σ=(1,−1) this specializes to {y1+y2, y2^2−1}, losing x1+x2. The claimed Step-2 success probability 1−h/|S| (h=0) is false; failure probability is at least 1/|S|. Proposition 3.4's bound is therefore unsupported, and the termination proof of Proposition 4.1, which relies on Step 2/3 successes, is affected (though the output-correctness part based on the final check is independent).
  2. [§5.1, Remark 5.1] The final correctness check in the shipped implementation uses a multi-modular Monte-Carlo Gröbner computation over Q for which no practical probability bound is known. Thus the implementation does not deliver the guarantee of Proposition 4.1. This is disclosed, which is commendable, but the abstract and introduction should state that the implementation is heuristic with respect to this final check; otherwise readers may assume the benchmarks are instances of the proven algorithm.
minor comments (5)
  1. [§3.3, Lemma 3.6] Typo: 'is and only if' should be 'if and only if'.
  2. [§2.7, Example 2.16] Notation 'OMSk(g1,g2)' should be 'OMS_{k(g1,g2)}'; also 'eOMS g1,g2' needs parentheses for clarity.
  3. [§2.5, Prop. 2.8 proof] Spelling: 'Schwarz-Zippel' should be 'Schwartz-Zippel'; 'Notedeg' is missing a space.
  4. [References] Reference [126] lists 'von zur Garthen'; the correct spelling is 'von zur Gathen'.
  5. [§3.3, Prop. 3.8] The assertion that b(a_i)≠0 makes normal-form computation and evaluation commute would benefit from a short proof; Lemma 3.3 as stated addresses Gröbner bases, not normal forms directly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the correctness chain is built on in-paper lemmas plus independent external theorems ([101], [124], [37]); self-citations are published inputs, not smuggled conclusions. Flagged concerns (Algorithm 6 Step 2 support recovery; Remark 5.1) are proof gaps, not circular reductions.

full rationale

The main correctness chain (Proposition 4.1, the paper's central claim) is not circular. The claim that (e)OMS Grobner-basis coefficients generate the field rests on Lemma 2.12 (external, [101]) and Lemma 2.15, proved in the paper by a normal-form/containment argument that uses only that eOMS_g has a solution y = x. The evaluation-interpolation pipeline (Algorithms 2, 3, 6 and Propositions 2.8-2.9, 3.4) builds on the external sparse-interpolation theorem of van der Hoeven-Lecerf [124] plus the in-paper Lemma 3.3; the error bounds are Schwartz-Zippel estimates on genuine error loci, and interpolated coefficients are checked at fresh random points, so there is no fitted-parameter-called-prediction pattern: the interpolant is not a 'prediction' statistically forced by the same data. Algorithm 4's membership test imports Theorem 2.19 from [37] and Lemma 2.20 from [68]; these are self-citations (co-author Pogudin) and are load-bearing for the verification steps, but each is a peer-reviewed published theorem whose stated assumptions do not include the present target result, so under the review rules they are independent evidence and do not raise the circularity score. Algorithm 7 is attributed to [69] but carries its own proof (Proposition 3.8). The quality claims in Section 6 are admittedly heuristic ('we have no single qualitative measure'), and the sorting criteria from Lemma 3.2 double as part of the quality assessment, which is mildly self-referential, but the paper also reports external measures (degrees, term counts, algebraic independence, comparison with [105]) and manual inspection. Two flagged concerns are genuine but are not circularity: (a) Algorithm 6 Step 2 reads monomial support from a single specialization G_sigma, while Lemma 3.3 guarantees equality of the sets of specialized polynomials, not preservation of the supports of non-leading coefficients; a non-leading coefficient such as x1+x2 in E = Q(x1+x2, x1x2) can vanish at sigma = (1,-1) with g(sigma) != 0, so the 1 - h/|S| bound for Step 2 in Proposition 3.4 is not supplied by the quoted lemma (a correctness gap, not a reduction to inputs); (b) Remark 5.1 concedes that the implemented final check over Q uses a multi-modular Monte-Carlo Grobner computation with no practical probability bound, so the shipped implementation may not deliver the theoretical guarantee. Neither concern makes the derivation equivalent to its own inputs by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim relies on established theorems about OMS ideals, sparse interpolation, and randomized membership testing, plus a hand-chosen degree parameter delta. There are no invented particles, mediators, or new mathematical entities, and no constants fitted to make the result work.

free parameters (1)
  • delta (degree bound for polynomial search) = 3 in benchmarks; 2 or 3 suggested in Algorithm 8 Step 4
    A hand-chosen small constant that determines which low-degree polynomials enter the candidate pool. It affects output quality and runtime but not the correctness guarantee.
assumptions (5)
  • standard math Every subfield E of k(x) is finitely generated over k (Lang, Algebra, Ch. VIII, Ex. 4).
    Used to justify representing E by a finite generator list in Definition 2.10 and throughout.
  • standard math OMS ideal coefficients of a reduced Gröbner basis generate E (Lemma 2.12, Lemma 2.15, building on [101]).
    Load-bearing for Algorithm 8 Step 2: it ensures the searched coefficients are a valid source of generators.
  • standard math Randomized subfield membership test correctness of [37, Theorem 3.3].
    Algorithm 4 uses this theorem; Steps 3, 6 and 7 of Algorithm 8 rely on the membership test's probabilistic correctness.
  • standard math Sparse rational function interpolation correctness of van der Hoeven and Lecerf [124, Theorem 2].
    Justifies Algorithms 2 and 3 and the probability bounds in Propositions 2.8 and 2.9.
  • standard math Jacobian and transcendence criteria from Ehrenborg-Rota [38, Proposition 2.4] and automorphism extension from Milne [95, Proposition 2.4(a)].
    Used in Lemma 2.20 and Proposition 2.21 to reduce membership testing to a zero-dimensional ideal computation.

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Cite this review

Pith. "Pith review of Simple generators of rational function fields." pith.science (2026). https://pith.science/paper/7UFCCY7Y

@misc{pith2026260210878,
  author       = {Pith},
  title        = {Pith review of: Simple generators of rational function fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UFCCY7Y}},
  note         = {Machine review of arXiv:2602.10878}
}
read the original abstract

Consider a subfield of the field of rational functions in several indeterminates. We present an algorithm that, given a set of generators of such a subfield, finds a simple generating set. We provide an implementation of the algorithm and show that it improves upon the state of the art both in efficiency and the quality of the results. Furthermore, we demonstrate the utility of simplified generators through several case studies from different application domains, such as structural parameter identifiability. The main algorithmic novelties include performing only partial Gr\"obner basis computation via sparse interpolation and efficient search for polynomials of a fixed degree in a subfield of the rational function field.

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Cited by 1 Pith paper

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Reviewed August 4, 2026 · model on record in the stance chip above.