{"id":"392aa84d-9978-404f-a18d-5474e9d76f25","arxiv_id":"2602.10878","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A new randomized algorithm computes simpler generators of rational function fields by interpolating only low-degree Gröbner coefficients and searching for low-degree polynomial elements.","lead":"This paper presents an algorithm that, given any set of rational-function generators of a field, returns a smaller, simpler generating set, using partial Gröbner-basis computations and sparse interpolation. It ships an open-source Julia implementation that massively shrinks generators arising in systems-biology identifiability, discrete-time models, and invariant theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 6's support recovery relies on a support-preservation property that Lemma 3.3 does not provide; Proposition 3.4's probability bound is therefore unsupported.","rationale":"The reader correctly flagged the genericity assumption behind Lemma 3.3 and the implementation gap in Remark 5.1. However, the most load-bearing issue is subtler: Lemma 3.3 guarantees equality of the specialized GB as a set of polynomials, but Algorithm 6 requires preservation of the set of monomials with nonzero coefficients. Since a non-leading coefficient can vanish at a specialization point that satisfies g(a)≠0, Step 2 can miss entire coefficients, and Proposition 3.4's stated probability bound is not valid. This is a proof gap in the central theoretical contribution, not merely an implementation concern. The gap is likely fixable by enlarging g to control all coefficient numerators, but as written the main theorem's proof is incomplete. The verdict remains CONDITIONAL (no change), because the algorithmic idea is plausible and the gap is repairable, but the paper needs revision to either prove the support-stability property or modify Algorithm 6 to handle vanishing non-leading coefficients.","tokens_in":58731,"tokens_out":16520,"duration_ms":164039,"concrete_test":"Take E=Q(x1+x2, x1x2). Compute the reduced Gröbner basis of eOMS over Q(x1,x2) and confirm the second polynomial contains y2 with coefficient −(x1+x2). Run Algorithm 6's Step 2 with σ=(1,−1); if the recovered support is {y2^2,1} (missing y2), the support-recovery step is unjustified. Broader check: instrument Algorithm 6 to compare, for each benchmark, the support recovered from one random σ with the true support obtained by computing the GB over Q(x); record any case where a nonzero coefficient vanishes at σ while g(σ)≠0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Algorithm 6, Step 2 recovers the support (monomials α_{i,j}) of the reduced Gröbner basis over k(x) from a single specialization σ, citing Lemma 3.3. That lemma only guarantees the specialized reduced basis equals the image of the true basis as a set of polynomials; it does not control non-leading coefficients. A non-leading coefficient c(x) can vanish at σ while g(σ)≠0, so its monomial is absent from the specialized basis and is never interpolated. Example: for E=Q(x1+x2, x1x2), eOMS has reduced GB {y1+y2−x1−x2, y2^2−(x1+x2)y2+x1x2}; at σ=(1,−1) (g≡1), the second polynomial becomes y2^2−1, losing the y2 term and its coefficient x1+x2. Thus Proposition 3.4's claim that Step 2 succeeds with probability at least 1−h/|S| (with h=deg g=0) is false; actual failure probability is at least ~1/|S|. This undermines the proof of Algorithm 8's correctness, since C^{≤d} may silently omit essential generators.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":59082,"tokens_out":28734,"duration_ms":320718,"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":[{"comment":"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).","section":"§3.2, Algorithm 6 / Prop. 3.4"},{"comment":"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.","section":"§5.1, Remark 5.1"}],"minor_comments":[{"comment":"Typo: 'is and only if' should be 'if and only if'.","section":"§3.3, Lemma 3.6"},{"comment":"Notation 'OMSk(g1,g2)' should be 'OMS_{k(g1,g2)}'; also 'eOMS g1,g2' needs parentheses for clarity.","section":"§2.7, Example 2.16"},{"comment":"Spelling: 'Schwarz-Zippel' should be 'Schwartz-Zippel'; 'Notedeg' is missing a space.","section":"§2.5, Prop. 2.8 proof"},{"comment":"Reference [126] lists 'von zur Garthen'; the correct spelling is 'von zur Gathen'.","section":"References"},{"comment":"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.","section":"§3.3, Prop. 3.8"}],"recommendation":"major_revision","confidential_remarks":"The central output-correctness bound in Prop. 4.1 is robust to the support-recovery issue because it relies on the final randomized check. The main problem is Proposition 3.4 and the termination proof. A fix should either enlarge the exceptional polynomial g in Lemma 3.3 to control all coefficients (not only leading ones) or modify Algorithm 6 to recover support from several specializations. I also suggest moving the Remark 5.1 caveat into the introduction. The paper is otherwise strong and the empirical results are compelling."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Uwe, here's the quick take: the paper is worth reading for the engineering and the benchmark story, but the main correctness proof has a hole that I think is real. Algorithm 6's Step 2 recovers the support of the reduced Gröbner basis by computing one specialization σ. The proof leans on Lemma 3.3, which only guarantees that the specialized basis equals the image of the true basis as a set of polynomials. It says nothing about non-leading coefficients. If a non-leading coefficient c(x) vanishes at σ, its monomial never appears in the specialized basis, and the algorithm never interpolates it. The stress-test example is concrete: for E=Q(x1+x2, x1x2), the reduced basis of eOMS is {y1+y2−x1−x2, y2^2−(x1+x2)y2+x1x2}. At σ=(1,−1), the second polynomial becomes y2^2−1, losing the y2 term. So C^{≤d} is missing x1+x2, which is essential to generate E. The probability bound in Proposition 3.4 therefore doesn't hold; the actual failure probability involves the zero sets of all non-leading coefficients, not just the g from Lemma 3.3. This undermines Algorithm 8's termination argument, since the field-equality check keeps failing and the missing coefficient never appears at higher d.\n\nWhat the paper does well: the idea of partial Gröbner extraction via sparse interpolation is genuinely new and well-motivated, the implementation is real (Julia, open source), the benchmark suite is large, and the case studies (identifiability, invariants, discrete-time observability) are compelling. The exposition is unusually careful about probability bounds, and Remark 5.1 openly admits the implementation's final check has no practical bound—that honesty is commendable. The 'simplicity' claim is heuristic, but they say so upfront.\n\nThe soft spots: besides the support gap, the paper's own Remark 5.1 is a real limitation: the shipped code may not deliver the stated probability guarantee. The comparison with [105] is a bit cherry-picked, but the across-the-board speedup seems genuine.\n\nWho benefits: people working on structural identifiability and invariant theory will get useful ideas and a tool, but they should not yet rely on the theoretical guarantee. A solid referee should engage because the approach is promising and the flaw is likely fixable (e.g., by recovering support from multiple specializations or by interpolating a superset of monomials). I would not cite the algorithm as proven yet, but I'd bring the paper to a reading group to dissect the proof.","headline":"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.","tokens_in":59487,"tokens_out":5910,"would_cite":false,"duration_ms":62915,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68W30","13P10","12F20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["rational function fields","simple generators","Gröbner bases","OMS ideals","sparse interpolation","structural identifiability","rational invariants","randomized algorithms"],"falsifier":"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.","tokens_in":58669,"feed_emoji":"🧮","tokens_out":6918,"duration_ms":72882,"temperature":0.7,"pith_summary":"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.","feed_headline":"One algorithm finds simple generators for rational-function subfields","feed_subtitle":"It shrinks huge parameter sets from biology and epidemiology to a handful of interpretable expressions.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Algorithm finds simple generators for rational function subfields","Simplify rational function subfields with one algorithm","New algorithm simplifies rational function field generators","One algorithm yields simple generators for rational subfields"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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","fun_headline_variants_meta":{"raw":{"variants":["Algorithm finds simple generators for rational function subfields","Simplify rational function subfields with one algorithm","New algorithm simplifies rational function field generators","One algorithm yields simple generators for rational subfields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00016,"raw_usage":{"total_tokens":1005,"prompt_tokens":619,"completion_tokens":386,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":363,"completion_tokens_details":{"reasoning_tokens":342}},"tokens_in":363,"tokens_out":386,"duration_ms":4784,"temperature":1.0,"reasoning_tokens":342,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:01:37.701534+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}