Valuative dimension and monomial orders

Gregor Kemper; Ihsen Yengui

arxiv: 1906.12067 · v1 · pith:SBBSDVY3new · submitted 2019-06-28 · 🧮 math.AC

Valuative dimension and monomial orders

Gregor Kemper , Ihsen Yengui This is my paper

Pith reviewed 2026-05-25 13:46 UTC · model grok-4.3

classification 🧮 math.AC

keywords valuative dimensionmonomial ordersgraded lexicographic orderKrull dimensionconstructive characterizationcommutative ringsrational monomial order

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The pith

Valuative dimension of a ring admits a constructive characterization via graded rational monomial orders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides a constructive test for the valuative dimension that mirrors Lombardi's test for Krull dimension but relies on graded reverse lexicographic monomial orders rather than plain lexicographic ones. A sympathetic reader would care because this supplies an algorithmic way to determine a dimension concept that refines the usual Krull dimension in the presence of valuations. The result applies to rings that support such monomial orders and includes some examples to illustrate the ideas. If the characterization holds, it extends constructive methods in commutative algebra to this valuative setting.

Core claim

The valuative dimension equals the maximum length of chains or sequences detected by the graded reverse lexicographic monomial order, or more generally by any graded rational monomial order, giving a direct constructive analogue to the known lex-order characterization of Krull dimension.

What carries the argument

Graded rational monomial order, which replaces the lexicographic order in the test for dimension and allows the constructive verification of valuative dimension.

If this is right

The valuative dimension becomes computable for polynomial rings and similar structures using standard monomial order algorithms.
Related results in the paper connect this to other dimension notions in algebra.
Examples demonstrate cases where the order detects the dimension correctly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach may extend to other non-Noetherian rings if they admit suitable orders.
Computational algebra systems could implement this test for practical dimension calculations.
Connections to valuation theory might be strengthened by this explicit order-based description.

Load-bearing premise

The rings under consideration admit a graded rational monomial order for which the stated constructive test correctly captures the valuative dimension.

What would settle it

A specific ring and monomial order where the maximum chain length under the order differs from the independently computed valuative dimension of the ring.

read the original abstract

The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's constructive characterization of the Krull dimension. While Lombardi's characterization uses the lexicographic monomial order, ours uses the graded (reverse) lexicographic order or, in fact, any graded rational monomial order. Apart from this, the paper contains some related results and some examples which readers may find illuminating.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Short note extends Lombardi's Krull-dimension characterization to valuative dimension via graded rational monomial orders.

read the letter

The main takeaway is that this note supplies a constructive characterization of valuative dimension that replaces the lexicographic order in Lombardi's Krull-dimension result with any graded rational monomial order, including graded reverse lex. That is the actual new piece on offer. The abstract frames it as a direct analogy, and the authors add some related results plus examples that could help readers see the test in action. For someone already working with monomial orders and constructive dimension theory, the flexibility in the choice of order looks like a practical improvement rather than a cosmetic change. The paper stays within its stated scope and does not claim broader reorganizations of the subject. The main soft spot is that only the abstract is available here, so the precise statement of the theorem, the hypotheses on the rings, and the proof steps cannot be inspected. Without those, it is impossible to confirm that the derivation avoids extra assumptions or that the test really works as claimed for the stated class of rings. The circularity burden appears low because the note positions itself as an extension of the existing Lombardi result rather than a self-contained derivation from scratch. This is a short, targeted note aimed at commutative algebraists who already know the prior constructive characterizations and care about monomial orders for explicit computations. A reader in that niche will get the point quickly and may find the examples useful. It is not a field-changing paper, but the result is new enough and the analogy is clean enough that it deserves a serious referee to check the details and the examples. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper provides a constructive characterization of the valuative dimension of a commutative ring, analogous to Lombardi's characterization of Krull dimension. While Lombardi's result relies on the lexicographic monomial order, the main theorem here uses the graded (reverse) lexicographic order or, more generally, any graded rational monomial order. The note also contains related results and illustrative examples.

Significance. If the main characterization holds, it supplies a new constructive test for valuative dimension that is directly computable via standard monomial-order machinery. This extends the scope of effective dimension theory in commutative algebra and may facilitate algorithmic applications in rings admitting suitable gradings. The analogy to Lombardi's result is a clear strength, as is the claim that the result works for any graded rational monomial order.

minor comments (2)

The abstract refers to 'the main result' and 'related results' without indicating section numbers; the manuscript should explicitly label the statement of the central theorem (e.g., Theorem 3.2 or similar) so readers can locate the precise claim immediately.
Examples are mentioned but not described; a brief indication of the rings or monomial orders used in the examples would help assess the scope of the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report correctly identifies the main result as a constructive characterization of valuative dimension via graded (reverse) lexicographic or any graded rational monomial order, in direct analogy to Lombardi's theorem for Krull dimension. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract frames the main result as a constructive characterization of valuative dimension that is explicitly analogous to an external result (Lombardi's characterization of Krull dimension) rather than derived from the paper's own inputs or prior self-citations. No equations, definitions, or load-bearing steps are supplied in the available text that would reduce a prediction or uniqueness claim to a fitted parameter or self-referential definition. The derivation is presented as building on independent prior work, making the paper self-contained against external benchmarks with no detectable circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5582 in / 934 out tokens · 30822 ms · 2026-05-25T13:46:27.928431+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 6. Let A be a ring, n a positive integer, and < a rational monomial preorder on n variables. (a) If dim_v(A) < n, then every sequence of n elements in A is dependent with respect to <. (b) If < is a graded monomial order, then the converse of (a) holds.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The main result ... uses the graded (reverse) lexicographic order or, in fact, any graded rational monomial order.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Pure Appl

Thierry Coquand, Space of valuations , Ann. Pure Appl. Logic 157 (2009), 97–109

work page 2009
[2]

Robert Gilmer, Multiplicative Ideal Theory, Marcel Dek ker Inc., New York, 1972

work page 1972
[3]

Paul Jaﬀard, Th´ eorie de la dimension dans les anneaux de polynomes, M´ emor. Sci. Math., Fasc. 146, Gauthier-Villars, Paris 1960

work page 1960
[4]

Gregor Kemper, A Course in Commutative Algebra, Springe r-Verlag, Berlin, Heidelberg 2011

work page 2011
[5]

Algebra 399 (2014), 782–800

Gregor Kemper, Ngo Viet Trung, Krull dimension and monomial orders , J. Algebra 399 (2014), 782–800

work page 2014
[6]

Gregor Kemper, Ngo Viet Trung, Nguyen Thi Van Anh, Toward a theory of monomial preorders , Math. Comp. 87 (2018), 2513-2537

work page 2018
[7]

Henri Lombardi, Dimension de Krull, Nullstellens¨ atze et ´ evaluation dynamique, Math. Z. 242 (2002), 23–46

work page 2002
[8]

Finite projective modules, Algebra and Applications, 20, Springer, Dordrecht 2015

Henri Lombardi, Claude Quitt´ e, Commutative algebra: c onstructive methods. Finite projective modules, Algebra and Applications, 20, Springer, Dordrecht 2015

work page 2015
[9]

Technische Universi ¨at M ¨unchen, Zentrum Mathematik - M11, Boltzmannstr

Hideyuki Matsumura, Commutative Ring Theory, Cambridg e University Press, Cambridge 1986. Technische Universi ¨at M ¨unchen, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Ga rching, Ger- many E-mail address : kemper@ma.tum.de Department of Mathematics, F aculty of Sciences of Sfax, Uni versity of Sfax, 3000 Sfax, Tunisia E-mail address : ihsen.yengui@f...

work page 1986

[1] [1]

Pure Appl

Thierry Coquand, Space of valuations , Ann. Pure Appl. Logic 157 (2009), 97–109

work page 2009

[2] [2]

Robert Gilmer, Multiplicative Ideal Theory, Marcel Dek ker Inc., New York, 1972

work page 1972

[3] [3]

Paul Jaﬀard, Th´ eorie de la dimension dans les anneaux de polynomes, M´ emor. Sci. Math., Fasc. 146, Gauthier-Villars, Paris 1960

work page 1960

[4] [4]

Gregor Kemper, A Course in Commutative Algebra, Springe r-Verlag, Berlin, Heidelberg 2011

work page 2011

[5] [5]

Algebra 399 (2014), 782–800

Gregor Kemper, Ngo Viet Trung, Krull dimension and monomial orders , J. Algebra 399 (2014), 782–800

work page 2014

[6] [6]

Gregor Kemper, Ngo Viet Trung, Nguyen Thi Van Anh, Toward a theory of monomial preorders , Math. Comp. 87 (2018), 2513-2537

work page 2018

[7] [7]

Henri Lombardi, Dimension de Krull, Nullstellens¨ atze et ´ evaluation dynamique, Math. Z. 242 (2002), 23–46

work page 2002

[8] [8]

Finite projective modules, Algebra and Applications, 20, Springer, Dordrecht 2015

Henri Lombardi, Claude Quitt´ e, Commutative algebra: c onstructive methods. Finite projective modules, Algebra and Applications, 20, Springer, Dordrecht 2015

work page 2015

[9] [9]

Technische Universi ¨at M ¨unchen, Zentrum Mathematik - M11, Boltzmannstr

Hideyuki Matsumura, Commutative Ring Theory, Cambridg e University Press, Cambridge 1986. Technische Universi ¨at M ¨unchen, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Ga rching, Ger- many E-mail address : kemper@ma.tum.de Department of Mathematics, F aculty of Sciences of Sfax, Uni versity of Sfax, 3000 Sfax, Tunisia E-mail address : ihsen.yengui@f...

work page 1986