Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings

Henri Lombardi (LMB); Ihsen Yengui; Stefan Neuwirth (LMB)

REVIEW 2 major objections 4 minor 26 references

Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that over a strongly discrete coherent base ring, the generalized Buchberger algorithm computes a Gröbner basis for a module exactly when the module of leading terms is finitely generated, and gives a constructive…

desk verdict The countable generation theorem and the Schreyer/Hilbert syzygy results are real, but the paper's headline Buchberger termination claim is delegated to a noetherian proof and is not actually proved in the stated generality. read the letter →

arxiv 2411.16460 v1 pith:OKQXCRR5 submitted 2024-11-25 math.AC

Stefan Neuwirth (LMB) , Henri Lombardi (LMB) , Ihsen Yengui This is my paper · ORCID

classification math.AC MSC 13D0213P1013C1013P2014Q20

keywords generalisedBuchbergeralgorithmsyzygytheoryfreeresolutionmonomialorderSchreyer'sconstructivemathematicscoherentringsGröbnerbases

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 works in constructive algebra and asks when Gröbner basis computations are possible for modules over polynomial rings whose coefficient ring is not a field. Its first main result is that, for a finitely generated submodule of a free module over $R[X_1,\ldots,X_n]$ with $R$ a discrete coherent ring, the module $MLT(M)$ of leading terms is countably generated, with an explicit algorithm enumerating generators via iterated S-lists. When $R$ is strongly discrete coherent, the paper proves Buchberger's criterion and algorithm remain valid: the generalized Buchberger algorithm terminates and yields a Gröbner basis if and only if $MLT(M)$ is finitely generated. It then proves Schreyer's algorithm computes a Gröbner basis for the first syzygy module under the induced order, giving a constructive Hilbert syzygy theorem with finite free resolutions of length at most $n+1$. The upshot is that the classical Buchberger and Schreyer methods extend to a broad class of non-noetherian coefficient rings.

What carries the argument

The central object is the iterated S-list $S^q(f_1,\ldots,f_p)$: starting with $S^0=(f_1,\ldots,f_p)$, each step appends the S-list $S(G)$ of the current list, whose elements are combinations $\sum_j S_{i,j}f_j$ formed from syzygies of the leading coefficients at each position level set $E$, lifted by multiplying each component by $\operatorname{lcm}(M_j)/M_j$. Since $R$ is coherent, each syzygy module of coefficients is finitely generated, so each S-list is finite; iterating provides the countable enumeration of leading terms. The rewriting algorithm based on these S-lists lowers the leading monomial of any linear combination with the same total leading monomial, which is what makes the Buchberger criterion and Schreyer's construction go through.

What would settle it

Take a nonarchimedean valuation domain $V$ with elements $a,b$ such that $a^q$ divides $b$ for every $q$, and run Algorithm 6.2 on $f_1=aX+1$, $f_2=b$ in $V[X]$. The theorem predicts the S-lists generate $\langle aX, b, b/a, \dots, b/a^q\rangle$ at stage $q$ and that the algorithm never terminates because $MLT$ is not finitely generated; if the run did terminate, or if the union of the stage submodules failed to equal the true $MLT$, the main equivalence would be false.

Watch

Extended reading notes

Core claim

Over a strongly discrete coherent base ring—a ring with decidable equality, membership tests with witnesses, and computable syzygy modules—the classical Buchberger and Schreyer constructions are not tied to noetherianness. For any $f_1,\ldots,f_p$, the iterated S-lists $S^q(f_1,\ldots,f_p)$ generate an increasing chain of leading-term submodules whose union is exactly $MLT(\langle f_1,\ldots,f_p\rangle)$, so the leading-term module is countably generated with an explicit enumeration. From this, the generalized Buchberger algorithm terminates exactly when $MLT$ is finitely generated, and the generalized Schreyer algorithm yields a Gröbner basis for the first syzygy module with respect to Schreyer's monomial order. Consequently every finitely generated submodule with finitely generated leading-term module has a finite free resolution of length at most $n+1$, constructively, with the leading terms of iterated syzygies successively independent of the indeterminates.

Load-bearing premise

The entire construction assumes the coefficient ring comes with algorithms that decide equality, test membership in finitely generated ideals with explicit witnesses, and compute finite generating sets for every syzygy module of a finite tuple; without these oracles the S-lists, division steps, and termination argument cannot be run.

Editorial extensions

If this is right

If $MLT(M)$ is finitely generated, the generalized Buchberger algorithm produces a Gröbner basis for $M$, so Gröbner bases become constructively available for all such modules over strongly discrete coherent rings.
If $MLT(M)$ is not finitely generated, the algorithm cannot terminate; the equivalence gives a computable witness for nontermination, since the increasing chain of leading-term submodules never stabilizes.
Schreyer's algorithm computes a Gröbner basis for the first syzygy module of any Gröbner basis, so syzygy modules inherit explicit Gröbner bases under the induced monomial order.
Every finitely generated submodule $U$ with $MLT(U)$ finitely generated admits a finite free resolution of length at most $n+1$, with the syzygy generators' leading terms independent of $X_n$, then $X_{n-1}$, and so on.
For Bézout and Prüfer coefficient rings, the S-list reduces to ordinary S-pairs and annihilator syzygies, recovering and unifying earlier syzygy theorems for such rings.

Reading between the lines

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

A reader may infer that when $MLT(M)$ is not finitely generated, the explicit enumeration still gives a semi-decision procedure: membership of a term in $MLT(M)$ can be confirmed by searching successively through $S^0, S^1, S^2, \dots$, although it cannot be refuted in finite time.
The same machinery suggests a uniform treatment of ideal membership and syzygy computation for 'division with remainder' coefficient rings beyond the strongly discrete case, as sketched in Remark 2.4.
If the constructive Hilbert syzygy bound $q\le n+1$ is independent of the base ring's complexity, then any coherent ring satisfying the algorithmic oracles yields resolutions of the same length as over a field, which is strong evidence that noetherianness is not the essential hypothesis for the theorem.

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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 short version. The paper has a genuinely new result—Theorem 4.11, the countable generation of the leading term module for any discrete coherent ring—and the Schreyer/Hilbert syzygy material in Section 7 is worked out in detail. But the main Buchberger claim (Theorem 6.1(2)) is not proved. The text says the proof 'parallels exactly' Adams–Loustaunau, yet their theorem is for strongly discrete coherent rings that are also noetherian. The paper supplies no argument that the coefficient ideals stabilize in the non-noetherian case even when MLT is finitely generated. So the iff in the abstract is unsupported as written.

What's good: the constructive syzygy machinery in Sections 4 and 5 is careful and genuinely useful. Proposition 4.4 and the iterated S-list construction are fully proved. The Schreyer order basis in Theorem 7.4 is proved in detail, and the Hilbert syzygy theorem in 7.5 follows from it cleanly—except that its first step assumes a Gröbner basis, which is supposed to come from Theorem 6.1(2). So the gap propagates.

On the soft spot: this is not a minor omission. It is the central termination theorem, and the paper's own Comment 4.12 explicitly notes that Adams–Loustaunau assume noetherianity, which makes the 'parallels exactly' handwave particularly suspicious. The proof is probably repairable—Dickson's lemma bounds the new leading monomials, and finite generation of MLT gives finite generation of each coefficient ideal, so the ascending chains should stabilize—but the authors need to actually write that out. As it stands, the paper claims more than it proves.

The reader's report underplays this; it calls the gap a 'completeness issue rather than a correctness flaw.' I think it is a correctness flaw in the current version because the main theorem statement is not established.

Who this is for: constructive and computational commutative algebraists interested in Gröbner bases over general rings. The countable generation result and the syzygy algorithms are worth a serious look. I would send this to a knowledgeable referee, but the verdict should be major revision, not acceptance. The authors should either prove Theorem 6.1(2) in the non-noetherian setting or soften the claim to match what is actually shown.

Referee Report

2 major / 4 minor

Summary. The paper develops a constructive theory of Gröbner bases for finitely generated submodules of a free module over a multivariate polynomial ring whose coefficient ring is a discrete coherent ring, or a strongly discrete coherent ring. It proves (Theorem 4.11) that the module of leading terms of such a module is countably generated and provides an explicit algorithm, via iterated S-lists, for producing a generating set. Under the stronger hypothesis that the base ring is strongly discrete coherent, it states a Buchberger criterion and a Buchberger algorithm (Theorem 6.1), claiming convergence if and only if the leading term module is finitely generated. It then gives Schreyer's algorithm for computing a Gröbner basis of the first syzygy module of a Gröbner basis (Theorem 7.4) and a constructive Hilbert syzygy theorem (Theorem 7.5) with resolution length at most n+1. Most technical proofs are provided; the main exception is Theorem 6.1, whose proof is only cited to Adams–Loustaunau.

Significance. If correct, these results extend Gröbner basis theory and Schreyer's syzygy method beyond the classical noetherian setting to a broad class of rings with strong algorithmic structure. The countable generation theorem and the iterated S-list algorithm are fully proved and are of independent interest, especially for non-noetherian valuation domains and other coherent rings. The constructive Hilbert syzygy theorem gives explicit finite free resolutions of length at most n+1 over such rings, which is a substantial generalization. The paper is also transparent in disclosing an error in prior work and correcting it. However, the central Buchberger algorithm theorem is not proved in the manuscript, so the main convergence claim is not established.

major comments (2)

[Theorem 6.1(2), Algorithm 6.2, Comment 4.12] The claim that Algorithm 6.2 converges whenever MLT(M) is finitely generated is one of the central results of the paper, but the proof is not given: the text says it 'parallels exactly' Adams–Loustaunau's Theorem 4.2.3. This is not sufficient, because the termination argument in Adams–Loustaunau's setting may use noetherianity: Comment 4.12 explicitly notes that their Theorem 4.2.8 assumes the base ring is noetherian, and noetherianity provides the ascending chain condition on submodules of a finitely generated module. For a non-noetherian strongly discrete coherent ring, a finitely generated module can contain infinite strictly ascending chains of submodules, so the chain of leading-term modules produced by Algorithm 6.2 need not stabilize merely because MLT(M) is finitely generated. The manuscript supplies no argument (e.g., a well-founded measure on the leading terms of the added remainders) showing that this particular chain must terminate. Therefore the 'if' direction of the claimed equivalence is not established.
[Theorem 6.1(1)] The Buchberger criterion is also stated without proof. The 'only if' direction follows from the definition of a Gröbner basis and the division algorithm, but the 'if' direction requires a substantial argument: one must show that the reduction of every S-polynomial to zero implies that every leading term of an element of M lies in <LT(G)>, using the fact that the S-list generates all syzygies of the leading terms (Proposition 4.4). This is non-obvious when the coefficient ring has nontrivial syzygies, especially for position-level sets E of size greater than 2. The citation to Adams–Loustaunau is not enough, as their proof may rely on hypotheses that are not present here; a self-contained proof or a precise indication of which steps are unchanged is needed.

minor comments (4)

[Algorithm 6.2] The for loop that removes elements from S and adds new elements to S within the same loop makes the control flow ambiguous; presenting it as a worklist algorithm with an explicit pending set would improve clarity.
[Example 4.7] The expression sE_1 = 1/3(-15,6) is not an equality in Z; the generator of Syz(6,15) should be given directly as (-5,2) (or (5,-2)) to avoid confusion about division in the base ring.
[Abstract and affiliations] There are typographical errors: 'multi variate' should be 'multivariate', and 'Bes ançon' should be 'Besançon'.
[Example 4.14(2)] The notation <LT(S0(f1,f2)> is missing a closing parenthesis; it should be <LT(S0(f1,f2))>.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are proved in the paper from the stated constructive hypotheses; self-citations are contextual or transparently corrected.

full rationale

The paper's derivation chain is not circular. Fundamental theorem 4.11 is proved directly from the definitions of iterated S-lists and the syzygy-of-terms construction, using only coherence/discreteness of the base ring and well-foundedness of monomial orders. Proposition 4.4 supplies the needed syzygy generation with a proof; Proposition 3.3, although attributed to the authors' prior work, is reproved in the text. Fundamental theorem 6.1 is stated as a parallel to Adams-Loustaunau 1994, but the algorithmic facts it relies on (S-list construction and division) are developed in Sections 4-5, and the if-direction is an application of Corollary 4.15 plus the division algorithm rather than a renaming of an input. Fundamental theorem 7.4 is proved by the Schreyer argument given in the paper, and Remark 7.6 openly corrects the authors' earlier formulation, which counts against concealment. The main potential weakness is that the proof of Theorem 6.1(2) is delegated to a source whose standing hypotheses include noetherianity; that is a completeness/correctness concern about the non-noetherian extension, not a circularity, since no fitted parameter or prior conclusion is being imported as the target result. Self-citations appear, but they are contextual (GLNY 2020, Yengui 2021a, 2024) or accompanied by proofs in this paper, so they are not load-bearing in the circularity sense.

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

The central results are conditional on the constructive hypotheses of discreteness, coherence, and strong discreteness, which are explicit in the paper and are not introduced ad hoc. No free parameters or new mathematical entities are introduced.

assumptions (5)

domain assumption R is discrete, i.e. equality in R is decidable.
Used from Definition 2.1 onward to identify leading terms and to run zero tests in the algorithms.
domain assumption R is coherent, i.e. the syzygy module of every tuple of elements is finitely generated with algorithms that generate and represent syzygies.
Provides the finite generating sets s^E_i for coefficient syzygies used in Definitions 4.1 and 4.9 and in Algorithms 5.4 to 5.7.
domain assumption R is strongly discrete, i.e. membership in finitely generated ideals is decidable with explicit witnesses.
Required by the division algorithm 5.2, which is the engine behind Buchberger's algorithm 6.2 and Schreyer's algorithm 7.2.
standard math Monomial orders on H^m_n are total, compatible with multiplication, and well-founded.
Definition 3.1 ensures every nonzero vector has a leading term and that rewriting processes in Propositions 4.4 and 4.6 terminate.
standard math The generalized Buchberger criterion in the strongly discrete coherent context is valid; its proof is delegated to Adams and Loustaunau 1994, Theorem 4.2.3.
Fundamental theorem 6.1(1) states this criterion and the paper does not carry out the proof, so the result rests on the cited reference.

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Cite this review

Pith. "Pith review of Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings." pith.science (2026). https://pith.science/paper/OKQXCRR5

@misc{pith2026241116460,
  author       = {Pith},
  title        = {Pith review of: Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKQXCRR5}},
  note         = {Machine review of arXiv:2411.16460}
}

read the original abstract

Let M be a finitely generated submodule of a free module over a multivariate polynomial ring with coefficients in a discrete coherent ring. We prove that its module MLT(M ) of leading terms is countably generated and provide an algorithm for computing explicitly a generating set. This result is also useful when MLT(M ) is not finitely generated. Suppose that the base ring is strongly discrete coherent. We provide a Buchberger-like algorithm and prove that it converges if, and only if, the module of leading terms is finitely generated. We also provide a constructive version of Hilbert's syzygy theorem by following Schreyer's method.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Adams and Philippe Loustaunau

William W. Adams and Philippe Loustaunau . An introduction to Gr\"obner bases. American Mathematical Society, Providence, 1994

work page 1994

[2] [2]

The trailing terms ideal

Faten Ben Amor and Ihsen Yengui. The trailing terms ideal. J. Algebra Appl., 20 0 (9): 0 2150153, 2021. doi:10.1142/S021949882150153X

work page doi:10.1142/s021949882150153x 2021

[3] [3]

Foundations of constructive analysis

Errett Bishop. Foundations of constructive analysis. McGraw-Hill, New York, 1967

1967

[4] [4]

Constructive analysis

Errett Bishop and Douglas Bridges. Constructive analysis. Grundlehren der ma\-the\-ma\-ti\-schen Wissenschaften, 279. Springer, Berlin, 1985

work page 1985

[5] [5]

german Ein A lgorithmus zum A uffinden der B asiselemente des R estklassenringes nach einem nulldimensionalen P olynomideal

Bruno Buchberger. german Ein A lgorithmus zum A uffinden der B asiselemente des R estklassenringes nach einem nulldimensionalen P olynomideal . Ph. D .\ thesis, german Mathematisches Institut, Universit \"a t Innsbruck , 1965. Translation by Michael P. Abramson: An algorithm for finding the basis elements of the residue class ring of a zero dimensional po...

doi:10.1016/j.jsc.2005.09.007 1965

[6] [6]

Cox, John Little, and Donal O'Shea

David A. Cox, John Little, and Donal O'Shea. Using algebraic geometry. Graduate Texts in Mathematics, 185. Springer, New York, second edition, 2005

work page 2005

[7] [7]

Modules sur les anneaux commutatifs: cours et exercices

Gema-Maria D\' az-Toca, Henri Lombardi, and Claude Quitt\'e. Modules sur les anneaux commutatifs: cours et exercices. Calvage & Mounet, Paris, 2014

work page 2014

[8] [8]

Dobbs and Ira J

David E. Dobbs and Ira J. Papick. When is D+M coherent? Proc. Amer. Math. Soc., 56: 0 51--54, 1976. doi:10.2307/2041572

work page doi:10.2307/2041572 1976