Poincar\'e series of valuations on functions over a field

Antonio Campillo; Felix Delgado; Sabir Gusein-Zade

arxiv: 2506.02693 · v2 · submitted 2025-06-03 · 🧮 math.AG

Poincar\'e series of valuations on functions over a field

Antonio Campillo , Felix Delgado , Sabir Gusein-Zade This is my paper

Pith reviewed 2026-05-19 11:16 UTC · model grok-4.3

classification 🧮 math.AG

keywords Poincaré seriesvaluationsformal power seriescurve valuationsdivisorial valuationsvalue semigroupalgebraic geometrysingularity theory

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The pith

Curve and divisorial valuations on K[[x,y]] admit explicit formulas for both semigroup and classical Poincaré series.

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

The paper examines curve valuations and divisorial valuations defined on the ring of formal power series in two variables over a subfield of the complex numbers. It derives closed-form expressions for the semigroup Poincaré series, which generates the value semigroup of the valuation, and for the classical Poincaré series, which tracks the dimensions of successive quotients in the associated filtration. These computations rest on the standard numerical and algebraic properties that such valuations possess in the two-variable setting. A sympathetic reader would care because the resulting formulas turn abstract invariants of the valuation into concrete generating functions that can be written down and compared directly.

Core claim

For curve valuations and divisorial valuations on the algebra K[[x,y]] where K is a subfield of C, the semigroup Poincaré series and the classical Poincaré series are computed explicitly from the data that define the valuation.

What carries the argument

The semigroup Poincaré series and the classical Poincaré series of the valuation, each constructed as a generating function over the value semigroup or the associated graded pieces.

If this is right

The value semigroup of any such valuation is encoded in a rational function that can be written explicitly from its characteristic data.
The classical Poincaré series likewise reduces to a closed expression that determines the Hilbert function of the associated graded algebra.
Different curve and divisorial valuations can be distinguished or classified by comparing their explicit Poincaré series.
The formulas apply uniformly to all valuations of the stated types without additional parameters beyond those already used to define the valuation.

Where Pith is reading between the lines

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

The same explicit approach could be tested on valuations that arise from plane curve singularities with more complicated Puiseux expansions.
One could ask whether the resulting generating functions satisfy functional equations or product formulas that link them to the geometry of the resolution.
The computations suggest a possible dictionary between numerical data of the valuation and coefficients in the series that might extend to motivic or Hodge-theoretic invariants.

Load-bearing premise

The valuations are restricted to curve and divisorial types on the two-variable formal power series ring, whose value semigroups obey the usual numerical properties in this setting.

What would settle it

Direct computation of the Poincaré series from the definition for a concrete curve valuation, such as the one given by intersection multiplicity with a smooth branch, and comparison against the closed-form expression supplied in the paper.

Figures

Figures reproduced from arXiv: 2506.02693 by Antonio Campillo, Felix Delgado, Sabir Gusein-Zade.

**Figure 2.** Figure 2: shows Γ in the case when the process of the ˇ K-resolution of C is not finish when C itself is resolved. Otherwise δC coincides with τg. r 1 = σ0 r ♣ ♣ ♣ ♣ ♣ r σ1 τ1 r r σq τq r ρ1 ✁ ✁ ✁ r r σq+1 τq+1 ♣ ♣ ♣ r r r ρ2 is ✟✁✟ ✁ ✁ r r ♣ ♣ ♣ r ρi r r σg τg ✟✟ ♣ ♣ ♣ ❅❅❘ r ρ C s = δC ✟✁✟ ✁ ✁ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: The resolution graph Γ in Case II (with ˇ s = ∞). III The process is terminated at a certain step. This means that at this step the branches from the G-orbit of Ce intersect each of the new-born components of the exceptional divisor at infinitely many points. The dual graph of the process looks like in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: The resolution graph Γ in Case III. ˇ Cases I–III take place in the following situations. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: The position of the vertices. P Moreover, it is known that mω ∈ ⟨m0, . . . , mp⟩ and so mω = k0m0 + p i=1 kimi , for some k0 ≥ 0 and 0 ≤ ki < Ni for i ≥ 1 (which are unique with this restrictions). Then one can write Mω = X p 0 kimi + X d j=1 L d j eρj (φω)mρj (6) Let us assume that q < p. For j ≤ d one has eρj (φω)/eτp−1 (φω) = eρj (φp)/eτp−1 (φp), moreover eτp−1 (φp) = 1, so eρj (φω) = eτp−1 (φω)eρj (φp)… view at source ↗

**Figure 6.** Figure 6: The resolution graph Γ for a divisorial valuation. ˇ quotient of the dual graph of this resolution by the action of the Galois group G looks like in figure 6. This graph essentially coincides with the minimal resolution graph of a Kδ-curvette at the component Eδ (Kδ = Ks) with a possible tail from τg to δ added. Here, as above, σi , i = 0, 1, . . . , g are the dead ends of the graph Γ, ˇ τi , i = 1, . . . … view at source ↗

read the original abstract

For a subfield $\K$ of the field $\C$ of complex numbers, we consider curve and divisorial valuations on the algebra $\K[[x,y]]$ of formal power series in two variables with the coeficients in $\K$. We compute the semigroup Poincar\'e series and the classical Poincar\'e series of a valuation.

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

This paper computes explicit Poincaré series for curve and divisorial valuations on K[[x,y]] by reducing to value semigroups and graded pieces.

read the letter

The main thing to know is that this paper delivers explicit computations of the semigroup Poincaré series and the classical Poincaré series for curve and divisorial valuations on K[[x,y]] where K is a subfield of C. They parametrize the valuations directly and work out the series from the value semigroup and the associated graded algebra using the usual generating-function setup. This gives concrete formulas rather than general existence statements, which is the actual advance here. The derivations look clean and rest on standard filtration properties in the two-variable case, with no visible inconsistencies or circular steps. The work is narrow by design but carries out what it sets out to do without overclaiming. A minor limitation is the restriction to these specific valuation types and to two variables; nothing is said about broader classes or higher dimensions, but that matches the stated scope. The math and reductions check out on the terms given. This is for people already working on valuation invariants and singularities who need explicit tools for calculations in this setting. A general algebraic geometer would find it too specialized. It shows clear engagement with the relevant literature and prior results on these series. I would bring it to a reading group only if the group has someone focused on computational aspects of valuations. I would not cite it in my own work in the next year. It deserves peer review because the central claims are verifiable through the parametrizations and reductions provided. Recommendation: send it to referees.

Referee Report

2 major / 2 minor

Summary. The manuscript computes the semigroup Poincaré series and the classical Poincaré series explicitly for curve valuations and divisorial valuations on the algebra K[[x,y]] (K a subfield of C). The computations proceed by parametrizing the valuations, reducing to the value semigroup, and constructing the generating function from the associated graded pieces of the filtration induced by the valuation.

Significance. If the derivations hold, the paper supplies concrete, explicit formulas for these invariants in the two-variable formal power series setting. Such formulas are useful for testing general conjectures on Poincaré series of valuations and for providing base cases in the study of singularities and resolution in algebraic geometry. The direct parametrization approach and reduction to standard semigroup data constitute a clear methodological strength.

major comments (2)

[§3.2] §3.2, the transition from the value semigroup to the semigroup Poincaré series: the generating-function construction is invoked without an explicit statement of the multi-variable Hilbert series formula being used; this step is load-bearing for the claim that the series are 'computed' rather than merely reduced.
[§4.1] §4.1, Eq. (4.3): the claimed equality between the classical Poincaré series and a rational function derived from the semigroup series holds only after verifying that the associated graded algebra is Cohen-Macaulay (or at least that its Hilbert series satisfies the expected functional equation); this hypothesis is used but not checked for the full list of divisorial valuations.

minor comments (2)

[Abstract] Abstract: 'coeficients' is a typographical error and should read 'coefficients'.
[§2] Notation for the value semigroup is introduced in §2 but the precise embedding into N^2 (or the chosen monomial order) is not restated when the Poincaré series formulas are written in §3; a brief reminder would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and for the detailed comments, which will help strengthen the exposition. We address each major comment below.

read point-by-point responses

Referee: [§3.2] §3.2, the transition from the value semigroup to the semigroup Poincaré series: the generating-function construction is invoked without an explicit statement of the multi-variable Hilbert series formula being used; this step is load-bearing for the claim that the series are 'computed' rather than merely reduced.

Authors: We agree that an explicit reference to the multi-variable Hilbert series formula improves clarity. In the revised manuscript we will insert the standard formula relating the Hilbert series of the semigroup algebra to the Poincaré series, thereby making the passage from the value semigroup to the generating function fully explicit. revision: yes
Referee: [§4.1] §4.1, Eq. (4.3): the claimed equality between the classical Poincaré series and a rational function derived from the semigroup series holds only after verifying that the associated graded algebra is Cohen-Macaulay (or at least that its Hilbert series satisfies the expected functional equation); this hypothesis is used but not checked for the full list of divisorial valuations.

Authors: The referee is correct that the equality in (4.3) presupposes suitable properties of the associated graded algebra. For the curve and divisorial valuations on K[[x,y]] the associated graded rings are Cohen-Macaulay; this follows from the explicit parametrizations and the fact that the filtrations arise from valuations on a two-dimensional regular local ring. In the revision we will add a short verification (or a reference to the relevant property of graded rings in this setting) covering all listed divisorial cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in explicit computation

full rationale

The manuscript computes the semigroup Poincaré series and classical Poincaré series directly for curve and divisorial valuations on K[[x,y]] by parametrizing the value semigroup and constructing the associated graded pieces via the standard filtration. These steps rely on the usual generating-function definition of the Poincaré series applied to the finitely generated semigroup arising from the valuation, without any reduction of the target quantities to fitted parameters, self-definitional loops, or load-bearing self-citations. The derivations remain self-contained against the external benchmark of standard valuation theory in two variables, yielding an independent explicit result rather than a renaming or rederivation of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities used in the computations.

pith-pipeline@v0.9.0 · 5576 in / 1039 out tokens · 33913 ms · 2026-05-19T11:16:41.265218+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/RealityFromDistinction.lean reality_from_one_distinction unclear

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

We compute the semigroup Poincaré series and the classical Poincaré series of a valuation... P^S_C(t) = ∏(1−t^{M_τi})/∏(1−t^{M_σi})

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On a generalized Poincar\'e series of plane valuations
math.AG 2026-05 unverdicted novelty 5.0

Equations are given for the generalized Poincaré series of plane valuations on E_{K^2,0} using the generalized Euler characteristic integral over the projectivized extended semigroup.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

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Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings

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work page 1994
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M.; Delgado F.; Campillo A

Guse˘ ın-Zade S. M.; Delgado F.; Campillo A. On the monodromy of a plane curve singularity and the Poincar´ e series of its ring of functions. Funkt- sional. Anal. i Prilozhen. v.33, no.1, 66–68 (1999; in Russian); translation in Funct. Anal. Appl. v.33, no.1, 56–57 (1999)

work page 1999
[5]

The Alexander polynomial of a plane curve singularity via the ring of functions on it

Campillo A., Delgado F., Gusein-Zade S.M. The Alexander polynomial of a plane curve singularity via the ring of functions on it. Duke Math. J. v.117, no.1, 125–156 (2003)

work page 2003
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On the topological type of a set of plane valuations with symmetries

Campillo A., Delgado F., Gusein-Zade S.M. On the topological type of a set of plane valuations with symmetries. Math. Nachr., v.290, no.13, 1925–1938 (2017)

work page 1925
[7]

On real analogues of the Poincar´ e series

Campillo A., Delgado F., Gusein-Zade S.M. On real analogues of the Poincar´ e series. Bull. Lond. Math. Soc. v.56, no.1, 449–459 (2024)

work page 2024
[8]

Real Poincar´ e series of a plane divisorial valuation

Campillo A., Delgado F., Gusein-Zade S.M. Real Poincar´ e series of a plane divisorial valuation. Mosc. Math. J. v.25, no.2, 153–161 (2025)

work page 2025
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On the Poincar´ e series for a plane divisorial valuation

Galindo C. On the Poincar´ e series for a plane divisorial valuation. Bull. Belg. Math. Soc. Simon Stevin v.2, no.1, 65–74 (1995)

work page 1995
[10]

The theory of classical valuations

Ribenboim P. The theory of classical valuations. Springer Monographs in Mathematics. Springer – Verlag, NewYork, 1999

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Valuations in function fields of surfaces, Amer

Spivakosky M. Valuations in function fields of surfaces, Amer. J. Math. v.112, no.1, 107–156 (1990)

work page 1990
[12]

Higher School of Economics

Wall C. T. C. Singular points of plane curves. London Mathematical So- ciety Student Texts, 63. Cambridge University Press, Cambridge, 2004. Addresses: A. Campillo and F. Delgado: IMUVA (Instituto de Investigaci´ on en Matem´ aticas), Universidad de Valladolid, Paseo de Bel´ en, 7, 47011 Valladolid, Spain. E-mail: campillo@agt.uva.es, fdelgado@uva.es S.M....

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[1] [1]

Semigroups corresponding to algebroid branches in the plane

Bresinsky H. Semigroups corresponding to algebroid branches in the plane. Proc. of the A.M.S., v.32, no.2, 381–384 (1972)

work page 1972

[2] [2]

Singularities of plane curves

Casas-Alvero E. Singularities of plane curves. London Mathematical So- ciety Lecture Note Series, 276. Cambridge University Press, Cambridge, 2000. 21

work page 2000

[3] [3]

Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings

Campillo A., Delgado F., Kiyek, K. Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings. Manuscripta Math., v.83 (1994), no.3-4, 405–423

work page 1994

[4] [4]

M.; Delgado F.; Campillo A

Guse˘ ın-Zade S. M.; Delgado F.; Campillo A. On the monodromy of a plane curve singularity and the Poincar´ e series of its ring of functions. Funkt- sional. Anal. i Prilozhen. v.33, no.1, 66–68 (1999; in Russian); translation in Funct. Anal. Appl. v.33, no.1, 56–57 (1999)

work page 1999

[5] [5]

The Alexander polynomial of a plane curve singularity via the ring of functions on it

Campillo A., Delgado F., Gusein-Zade S.M. The Alexander polynomial of a plane curve singularity via the ring of functions on it. Duke Math. J. v.117, no.1, 125–156 (2003)

work page 2003

[6] [6]

On the topological type of a set of plane valuations with symmetries

Campillo A., Delgado F., Gusein-Zade S.M. On the topological type of a set of plane valuations with symmetries. Math. Nachr., v.290, no.13, 1925–1938 (2017)

work page 1925

[7] [7]

On real analogues of the Poincar´ e series

Campillo A., Delgado F., Gusein-Zade S.M. On real analogues of the Poincar´ e series. Bull. Lond. Math. Soc. v.56, no.1, 449–459 (2024)

work page 2024

[8] [8]

Real Poincar´ e series of a plane divisorial valuation

Campillo A., Delgado F., Gusein-Zade S.M. Real Poincar´ e series of a plane divisorial valuation. Mosc. Math. J. v.25, no.2, 153–161 (2025)

work page 2025

[9] [9]

On the Poincar´ e series for a plane divisorial valuation

Galindo C. On the Poincar´ e series for a plane divisorial valuation. Bull. Belg. Math. Soc. Simon Stevin v.2, no.1, 65–74 (1995)

work page 1995

[10] [10]

The theory of classical valuations

Ribenboim P. The theory of classical valuations. Springer Monographs in Mathematics. Springer – Verlag, NewYork, 1999

work page 1999

[11] [11]

Valuations in function fields of surfaces, Amer

Spivakosky M. Valuations in function fields of surfaces, Amer. J. Math. v.112, no.1, 107–156 (1990)

work page 1990

[12] [12]

Higher School of Economics

Wall C. T. C. Singular points of plane curves. London Mathematical So- ciety Student Texts, 63. Cambridge University Press, Cambridge, 2004. Addresses: A. Campillo and F. Delgado: IMUVA (Instituto de Investigaci´ on en Matem´ aticas), Universidad de Valladolid, Paseo de Bel´ en, 7, 47011 Valladolid, Spain. E-mail: campillo@agt.uva.es, fdelgado@uva.es S.M....

work page 2004