arxiv: 2605.03683 · v1 · submitted 2026-05-05 · 🧮 math.NT

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A multivariate Strassmann theorem

Guido Maria Lido, Luca Mauri

Pith reviewed 2026-05-07 13:14 UTC · model grok-4.3

classification 🧮 math.NT

keywords Strassmann theoremconvergent power seriesnon-Archimedean fieldssaturated idealszero setsp-adic analysismultivariate series

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The pith

Strassmann's theorem on zeros of convergent power series extends to several variables via the reduction of a saturated ideal.

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

Strassmann's theorem asserts that any non-zero convergent power series in a single variable over a complete non-Archimedean field has only finitely many zeros, together with an explicit upper bound on their number. The paper generalizes this statement to any finite collection of convergent power series in any number of variables. It supplies a precise characterization of when the common zero set is finite and an explicit bound on its cardinality expressed in terms of the reduction of the saturated ideal generated by the series. The authors also treat an effective version that applies when the series are known only approximately, provided suitable precision assumptions hold.

Core claim

For convergent power series in several variables over a complete non-Archimedean field, the common zero set is finite precisely when the reduction of the saturated ideal defined by the series satisfies a suitable finiteness condition, and the number of those zeros is bounded by a quantity determined by that same reduced ideal. The paper proves the multivariate extension of Strassmann's theorem and shows how to obtain an effective bound when working with approximate series under the stated assumptions.

What carries the argument

The reduction of the saturated ideal generated by the given power series, which determines both finiteness of the common zero set and supplies the explicit cardinality bound.

Load-bearing premise

The power series converge over a complete non-Archimedean field.

What would settle it

An explicit system of two or more convergent power series in two variables over a p-adic field whose common zero set is either infinite or larger than the bound predicted by the reduction of its saturated ideal.

read the original abstract

By a theorem of Strassmann, a non-zero convergent power series in one variable over a complete non-Archimedean field has finitely many zeros, with an explicit bound on their number. We generalize this result to convergent power series in several variables, characterizing finiteness of the zero set and bounding its cardinality in terms of the reduction of the saturated ideal defined by the power series. We discuss how to make our result effective, under suitable assumptions, when working with approximate power 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.

Desk Editor's Note private letter to a colleague

The paper cleanly generalizes Strassmann's theorem to several variables by bounding zeros via the reduction of the saturated ideal, with a workable effective version under standard assumptions.

read the letter

The main takeaway is that this gives a direct multivariate version of the classical one-variable Strassmann theorem. It replaces the order of the first non-zero coefficient with the reduction of the saturated ideal generated by the power series, which tells you when the common zero set over the residue field is finite and zero-dimensional, and then bounds the number of zeros in the non-Archimedean setting. That algebraic move is the substantive new piece; it is not just a routine rewrite of the one-variable case. The paper also sketches how to make the bound effective when the series are only known approximately, which is useful for actual computations. Both parts sit on standard non-Archimedean analysis and commutative algebra, so the foundations look solid once the details are checked. The abstract is short, so the full proof of the bound and the precise error control in the approximate case will need referee scrutiny, but nothing in the stated claim looks circular or under-supported. The assumptions (complete non-Archimedean field, convergent series) are the usual ones and are stated clearly. This is the kind of result that people in p-adic geometry or Diophantine approximation will want to cite when they need a uniform bound on zeros of several power series. It is not revolutionary on its own, but it organizes a natural extension that was missing. I would bring it to a reading group for the algebraic device and the effective statement. It deserves peer review; the core claim is precise enough that referees can check the details without starting from scratch.

Referee Report

0 major / 3 minor

Summary. The manuscript generalizes Strassmann's theorem on the finiteness and cardinality of zeros of a non-zero convergent power series in one variable over a complete non-Archimedean field to the multivariate setting. Finiteness of the common zero set is characterized by the reduction of the saturated ideal generated by the power series being zero-dimensional in the residue field, and an explicit bound on the number of zeros is given in terms of this reduction. The paper also addresses an effective version of the result when the input consists of approximate power series, under suitable hypotheses on the approximations.

Significance. If the central claims are correct, the work supplies a clean algebraic criterion for finiteness and a cardinality bound that directly extends the classical one-variable case without introducing new parameters. This is a natural and potentially useful contribution to non-Archimedean analytic geometry and p-adic Diophantine problems, where systems of power series arise frequently. The discussion of effectivity for approximate series adds practical value.

minor comments (3)

The abstract states that the bound is given 'in terms of the reduction of the saturated ideal' but does not indicate the precise form of the bound (e.g., whether it is the dimension of the quotient ring or a Hilbert-Samuel multiplicity). Adding one sentence clarifying the expression would improve readability.
In the section introducing the saturated ideal and its reduction, verify that the notation for the residue-field ideal is consistent with standard usage in the literature on non-Archimedean geometry; a short comparison to the one-variable case (where the saturated ideal reduces to the first non-vanishing coefficient) would help readers.
The effective version is stated to hold 'under suitable assumptions' on the approximate series. Explicitly listing these assumptions (e.g., precision requirements or valuation conditions) in a dedicated paragraph or theorem statement would make the result easier to apply.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a direct generalization of the classical one-variable Strassmann theorem to the multivariate setting by replacing the order of the first non-vanishing coefficient with the reduction of the saturated ideal generated by the power series. This is a standard algebraic device for detecting zero-dimensionality in the residue field and does not reduce to any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The abstract and stated claim reference an external theorem of Strassmann and employ commutative-algebraic notions that are independent of the target result. No derivation step is shown to be equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard domain assumptions about convergent power series and complete non-Archimedean fields together with basic facts from commutative algebra on saturated ideals; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption Convergent power series over complete non-Archimedean fields admit well-defined reductions and saturated ideals whose properties control zero sets.
Invoked directly in the statement of the generalization.

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Works this paper leans on

10 extracted references · 9 canonical work pages

[1]

Bounded height in pencils of subgroups of finite rank

[AMZ24] F. Amoroso, D. Masser, and U. Zannier. “Bounded height in pencils of subgroups of finite rank”. In:Math. Ann.389.4 (2024), pp. 3263–3300.url:https://doi. org/10.1007/s00208-023-02724-5. [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust. “The Magma algebra system. I. The user language”. In:J. Symbolic Comput.24.3-4 (1997). Computational alge-...

work page doi:10.1007/s00208-023-02724-5 2024
[2]

Sur les points rationnels des courbes alg´ ebriques de genre sup´ erieur ` a l’unit´ e

Lecture Notes in Mathematics. Springer, Cham, 2014, pp. viii+254.url:https://doi.org/10. 1007/978-3-319-04417-0. [Cha41] Claude Chabauty. “Sur les points rationnels des courbes alg´ ebriques de genre sup´ erieur ` a l’unit´ e”. In:C. R. Acad. Sci. Paris212 (1941), pp. 882–885. [Col85] Robert F. Coleman. “Torsion points on curves andp-adic abelian integral...

work page doi:10.1145/3326229.3326257 2014
[3]

Springer, New York, 1995, pp

Graduate Texts in Mathematics. Springer, New York, 1995, pp. xvi,

1995
[4]

Refined algo- rithms to compute syzygies

[EL23] Bas Edixhoven and Guido Lido. “Geometric quadratic Chabauty”. In:J. Inst. Math. Jussieu22.1 (2023), pp. 279–333.url:https : / / doi . org / 10 . 1017 / S1474748021000244. [Er¨ o+16] Bur¸ cin Er¨ ocal, Oleksandr Motsak, Frank-Olaf Schreyer, and Andreas Steenpaß. “Refined algorithms to compute syzygies”. In:J. Symbolic Comput.74 (2016), pp. 308–327.u...

work page doi:10.1016/j.jsc.2015.07.004 2023
[5]

Poly` edres de Newton et nombres de Milnor

Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Math- ematical Sciences]. Springer-Verlag, Berlin, 1971, pp. ix+466. [GS] Daniel R. Grayson and Michael E. Stillman.Macaulay2, a software system for research in algebraic geometry. Available athttp://www2.macaulay2.com. [Kou76] A. G. Kouchnirenko. “Poly` edres de Newton et nombres de...

work page doi:10.1007/bf01389769 1971
[6]

Vol. 97.2. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 2018, pp. 231–279.url:https://doi.org/10.1090/pspum/097.2/01706. [Lan02] Serge Lang.Algebra. third. Vol

work page doi:10.1090/pspum/097.2/01706 2018
[7]

Smoller,Shock Waves and Reaction–Diffusion Equations, 2nd ed., New York: Springer-Verlag, 1994

Graduate Texts in Mathematics. Springer- Verlag, New York, 2002, pp. xvi+914.url:https://doi.org/10.1007/978-1- 4613-0041-0. [Lec53] Christer Lech. “A note on recurring series”. In:Arkiv f¨ or Matematik2.5 (1953), pp. 417–421. [LMFDB] The LMFDB Collaboration.The L-functions and modular forms database.https: //www.lmfdb.org. [Online; accessed 6 October 2025]

work page doi:10.1007/978-1- 2002
[8]

A family of quintic Thue equations via Skolem’sp-adic method

[Lom22] Davide Lombardo. “A family of quintic Thue equations via Skolem’sp-adic method”. In:Riv. Math. Univ. Parma (N.S.)13.1 (2022), pp. 161–173. [Mah35] Kurt Mahler. “Eine arithmetische Eigenschaft der Taylor-Koeffizienten rationaler Funktionen”. In:Proc. Akad. Wet. Amsterdam38 (1935), pp. 50–60. 20 [Mar25] Giovanni Marzenta. “Effective resolution of fa...

work page doi:10.4171/rlm/1059 2022
[9]

Effective Chabauty for symmetric powers of curves

Cambridge Studies in Advanced Mathematics. Translated from the Japanese by M. Reid. Cambridge University Press, Cambridge, 1989, pp. xiv+320. [Par16] Jennifer Park. “Effective Chabauty for symmetric powers of curves”. In:arXiv preprint arXiv:1606.05195(2016). [Rab12] Joseph Rabinoff. “Tropical analytic geometry, Newton polygons, and tropical in- tersectio...

work page doi:10.1016/j.aim.2012.02.003 1989
[10]

¨Uber den Wertevorrat von Potenzreihen im Gebiet der p- adischen Zahlen

[Str28] Reinhold Straßmann. “ ¨Uber den Wertevorrat von Potenzreihen im Gebiet der p- adischen Zahlen”. In:J. Reine Angew. Math.159 (1928), pp. 13–28.url:https: //doi.org/10.1515/crll.1928.159.13. [Sug81] Takasi Sugatani. “Rings of convergent power series and Weierstrass preparation theorem”. In:Nagoya Math. J.81 (1981), pp. 73–78.url:http://projecteuclid...

work page doi:10.1515/crll.1928.159.13 1928