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arxiv: 2605.05998 · v1 · submitted 2026-05-07 · 🧮 math.AG

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On a generalized Poincar\'e series of plane valuations

F. Delgado , S.M. Gusein-Zade
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Pith reviewed 2026-05-08 06:12 UTC · model grok-4.3

classification 🧮 math.AG
keywords mathbbgeneralizedpoincarseriesmathcalplanevaluationsalgebra
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The pith

The generalized motivic Poincaré series for collections of plane valuations on E_{K^2,0} admit explicit equations.

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

This paper extends two earlier definitions of motivic generalized Poincaré series for plane valuations on the full algebra of complex holomorphic germs to the restricted algebra E_{K^2,0} whose Taylor coefficients lie in a fixed subfield K of the complexes. The natural version in this setting is defined by integrating the generalized Euler characteristic over the projectivized extended semigroup of the collection. Equations are derived that determine this series, building on prior explicit formulas that existed only for the ordinary Poincaré series of single curve or divisorial valuations. A sympathetic reader would care because the series encode algebraic and topological data attached to the valuations, which appear in the study of plane curve singularities and their resolutions.

Core claim

For a collection of plane valuations on the algebra E_{K^2,0}, the generalized Poincaré series defined as the integral of the generalized Euler characteristic over the projectivization of the extended semigroup satisfies explicit equations that permit its computation.

What carries the argument

The integral of the generalized Euler characteristic over the projectivized extended semigroup of the collection of valuations.

Load-bearing premise

The definition of the generalized Poincaré series via the generalized Euler characteristic integral over the projectivized extended semigroup extends directly to E_{K^2,0} and admits explicit closed-form or recursive equations.

What would settle it

For a concrete choice of subfield K and a small collection of explicit plane valuations on E_{K^2,0}, compute the integral definition directly and compare the result with the value obtained from the proposed equations.

Figures

Figures reproduced from arXiv: 2605.05998 by F. Delgado, S.M. Gusein-Zade.

Figure 1
Figure 1. Figure 1: The graph Γ of the minimal G-resolution. The Galois group G acts on the graph Γ and the quotient Γ by the action ˇ looks like in Figures 2 (in the finite case) or 3 (in the infinite one). The graph r 1 = σ0 r ♣ ♣ ♣ ♣ ♣ r σ1 τ1 r r σq τq r ρ1 ✁ ✁ ✁ r r σq+1 τq+1 ♣ ♣ ♣ r r ✟✁✟ ✁ ✁ r ρ2 r r ♣ ♣ ♣ r ρi r r σg τg ✟✟ ♣ ♣ ♣ ❅❅❘ r ρ C s = δC ✟✁✟ ✁ ✁ view at source ↗
Figure 2
Figure 2. Figure 2: The graph Γ in the finite case. ˇ r 1 = σ0 r ♣ ♣ ♣ ♣ ♣ r σ1 τ1 r r σq τq is r ρ1 ✁ ✁ ✁ r r σq+1 τq+1 ♣ ♣ ♣ r r r ρj ✟✁✟ ✁ ✁ r r σg τg ♣ ♣ ♣ ♣ r✟✟ ♣ ♣ ♣ r ρk ✟✁✟ ✁ ✁ ♣ ♣ ♣ ♣ view at source ↗
Figure 3
Figure 3. Figure 3: The graph Γ in the infinite case. ˇ Γ contains a subgraph which geometrically coincides with the dual graph of a usual resolution of the curve C (or of the process of its resolution): the lower part of Γ on view at source ↗
Figure 4
Figure 4. Figure 4: The graph Γ for a divisorial valuation. ˇ dual graph of the minimal G-invariant resolution looks as on view at source ↗
Figure 5
Figure 5. Figure 5: Resolution graphs of the divisorial valuations under consideration view at source ↗
read the original abstract

Earlier, there were defined two generalized (``motivic'') versions of the Poincar\'e series of a collection of plane valuations on the algebra ${\mathcal O}_{{\mathbb C}^2,0}$ of germs of holomorphic functions in two variables. One of them was defined as an integral with respect to the generalized Euler characteristic over the projectivization of the extended semigroup of the collection. One has a natural version of it for valuations on the algebra ${\mathcal E}_{{\mathbb K}^2,0}$ of germs of holomorphic functions in two variables whose Taylor coefficients are from a fixed subfield ${\mathbb K}$ of the field ${\mathbb C}$ of complex numbers. In this setting the usual Poincar\'e series were computed for one plane curve or divisorial valuation on ${\mathcal E}_{{\mathbb K}^2,0}$. We give equations for the corresponding generalized Poincar\'e series.

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.

Referee Report

0 major / 2 minor

Summary. The paper extends two previously defined generalized (motivic) versions of the Poincaré series for collections of plane valuations on the algebra O_{C^2,0} to the algebra E_{K^2,0} of germs of holomorphic functions with Taylor coefficients in a fixed subfield K of C. One version is defined via an integral with respect to the generalized Euler characteristic over the projectivization of the extended semigroup. Building on prior computations of the usual Poincaré series for single plane curve or divisorial valuations on E_{K^2,0}, the authors provide explicit equations for the corresponding generalized series in this setting.

Significance. If the claimed equations hold and are explicit, the work extends motivic invariants and Poincaré series computations to a coefficient-restricted algebraic setting, potentially enabling comparisons between the full complex case and subfield versions. The reliance on combinatorial data from the extended semigroup is a positive feature if the derivations avoid ad-hoc parameters, though no machine-checked proofs or reproducible code are mentioned.

minor comments (2)
  1. [Abstract] Abstract: the claim that the generalized series 'admits explicit closed-form or recursive equations' is stated without indicating which form the new equations take or how they differ from the single-valuation cases already computed.
  2. [Abstract] The abstract introduces E_{K^2,0} and the integral definition without a brief reminder of the generalized Euler characteristic or projectivized extended semigroup; adding one sentence would improve accessibility for readers familiar only with the O_{C^2,0} case.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for their summary of our manuscript and for recognizing its potential to extend motivic invariants to the coefficient-restricted setting. We address the key points from the report below.

read point-by-point responses

  1. Referee: The paper extends two previously defined generalized (motivic) versions of the Poincaré series for collections of plane valuations on the algebra O_{C^2,0} to the algebra E_{K^2,0} of germs of holomorphic functions with Taylor coefficients in a fixed subfield K of C.

    Authors: This correctly captures the scope of the work. The two generalized versions are adapted in a natural way to E_{K^2,0} by restricting the Taylor coefficients while preserving the integral definition over the projectivized extended semigroup. revision: no

  2. Referee: One version is defined via an integral with respect to the generalized Euler characteristic over the projectivization of the extended semigroup.

    Authors: Yes, this is the version for which we derive the explicit formula in the E_{K^2,0} setting, following the same integral construction used in the complex case. revision: no

  3. Referee: Building on prior computations of the usual Poincaré series for single plane curve or divisorial valuations on E_{K^2,0}, the authors provide explicit equations for the corresponding generalized series in this setting.

    Authors: We rely on the already-established explicit formulas for the ordinary Poincaré series in the E_{K^2,0} case and lift them to the generalized versions via the integral with respect to the generalized Euler characteristic. The resulting equations are expressed directly in terms of the combinatorial data of the extended semigroup. revision: no

  4. Referee: If the claimed equations hold and are explicit, the work extends motivic invariants and Poincaré series computations to a coefficient-restricted algebraic setting, potentially enabling comparisons between the full complex case and subfield versions. The reliance on combinatorial data from the extended semigroup is a positive feature if the derivations avoid ad-hoc parameters, though no machine-checked proofs or reproducible code are mentioned.

    Authors: The derivations in the manuscript are fully explicit and combinatorial; they use only the structure of the extended semigroup and the definition of the generalized Euler characteristic, with no ad-hoc parameters introduced. As this is a theoretical paper in algebraic geometry, we supply complete mathematical proofs rather than machine-checked code or computational verification. The arguments can be checked directly from the text. revision: no

Circularity Check

0 steps flagged

No significant circularity; extension follows from prior combinatorial definitions without reduction to fit or self-citation

full rationale

The paper defines a natural extension of the generalized Poincaré series (via generalized Euler characteristic integral over the projectivized extended semigroup) to the algebra E_{K^2,0} and states that it admits explicit equations, building directly on earlier definitions for O_{C^2,0} and prior computations of the usual series for single valuations. No equation or claim reduces by construction to a fitted input, no uniqueness theorem is imported from the authors' own prior work to force the result, and the central derivation remains independent of any self-citation chain. The construction is presented as combinatorial and direct, with no evidence of self-definitional loops or renamed empirical patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior definitions of the two generalized Poincaré series and on the existence of a well-defined generalized Euler characteristic on the projectivized extended semigroup. No new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The generalized Euler characteristic is defined on the projectivization of the extended semigroup of a collection of plane valuations.
    Invoked in the abstract as the basis for one of the generalized versions.
  • domain assumption The natural version of the generalized series extends from O_{C^2,0} to E_{K^2,0}.
    Stated as the setting in which the new equations are given.

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Reference graph

Works this paper leans on

15 extracted references · 1 canonical work pages · 1 internal anchor

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