Some factorization results for formal power series

Jitender Singh; Rishu Garg

arxiv: 2501.05375 · v2 · pith:U2B32QW7new · submitted 2025-01-09 · 🧮 math.NT · math.AC· math.AG

Some factorization results for formal power series

Rishu Garg , Jitender Singh This is my paper

Pith reviewed 2026-05-23 05:43 UTC · model grok-4.3

classification 🧮 math.NT math.ACmath.AG

keywords formal power seriesfactorizationirreducibility criteriaNewton polygonsDumas criterionprincipal ideal domainsdiscrete valuation domains

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The pith

Formal power series over principal ideal domains admit sharp bounds on irreducible factors that yield irreducibility criteria.

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

The paper derives factorization results for formal power series over principal ideal domains that give sharp upper bounds on the number of irreducible factors. These bounds are obtained from the prime factorization of the constant term together with data on selected higher-order coefficients. The bounds directly produce tests that decide irreducibility of a given series. The same approach extends the classical Dumas criterion from polynomials to formal power series over discrete valuation domains through the theory of Newton polygons.

Core claim

Using the prime factorization of the constant term up to a unit and information about some higher order terms, formal power series over principal ideal domains admit factorizations with sharp bounds on the number of irreducible factors; these bounds furnish irreducibility criteria. Further, the theory of Newton polygons extends the classical Dumas irreducibility criterion to formal power series over discrete valuation domains and thereby yields several irreducibility criteria.

What carries the argument

Newton polygons for formal power series over discrete valuation domains, which translate coefficient data into factorization bounds and irreducibility tests.

If this is right

The number of irreducible factors of a formal power series over a principal ideal domain is bounded in terms of the factorization of its constant term.
Several concrete irreducibility criteria for formal power series follow directly from the factorization bounds.
The classical Dumas criterion applies verbatim to formal power series once Newton polygons are defined in the power-series setting.
The bounds remain valid over any principal ideal domain and any discrete valuation domain.

Where Pith is reading between the lines

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

The same Newton-polygon technique could be checked on explicit series such as the exponential or logarithm to see whether the new criteria detect known irreducible cases.
The factorization bounds may suggest analogous results for power series over rings that are not principal ideal domains.
Implementation of the criteria in a computer algebra system would allow systematic testing of large families of series.

Load-bearing premise

The construction and basic properties of Newton polygons extend from polynomials to formal power series over discrete valuation domains with no further restrictions.

What would settle it

An explicit formal power series over a discrete valuation domain that factors into more irreducible factors than the Newton-polygon bound predicts, or that factors nontrivially despite satisfying the extended Dumas criterion.

read the original abstract

In this paper, we obtain some factorization results on formal power series over principle ideal domains with sharp bounds on number of irreducible factors. These factorization results correspondingly lead to irreducibility criteria for formal power series. The information about prime factorization of the constant term up to a unit and that of some higher order terms is utilized for the purpose. Further, using theory of Newton polygons for power series, we extend the classical Dumas irreducibility criterion to formal power series over discrete valuation domains, which in particular, yields several irreducibility criteria.

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 extends Dumas' criterion to formal power series over DVRs and gives factorization bounds over PIDs, but the Newton polygon step for infinite series is not obviously justified.

read the letter

The core claim is an extension of the classical Dumas irreducibility criterion to formal power series over discrete valuation domains, plus some factorization results over PIDs that bound the number of irreducible factors using the constant term and selected higher coefficients. That extension is presented as the main new content, building on the cited prior work on Newton polygons and Dumas' original polynomial result. The factorization bounds appear to be a direct application of valuation data rather than a new framework. Credit is due for trying to carry the polygon method into the power series setting, which is a natural direction if it works cleanly. The abstract does not supply proofs or explicit statements of any extra hypotheses, so the soundness cannot be checked from what is given. The stress-test point about infinite support is worth taking seriously: Dumas' original construction uses a finite set of points, and for a general series the lower convex hull is only well-defined if the valuations tend to infinity fast enough or the series satisfies a Weierstrass-type condition. The abstract invokes the theory for power series without mentioning such a restriction, so the extension rests on an implicit assumption that may or may not hold in the full argument. If the paper restricts to series where the polygon is finite or proves the hull behaves as needed, that would close the gap; otherwise the claim is overstated. This is specialized commutative algebra and number theory material. A reader already working on irreducibility criteria or Newton polygons for non-polynomial rings could extract the bounds and the claimed criteria if the proofs are present and correct. It is not a broad reorganization of the subject. The work deserves a serious referee to verify whether the Newton polygon extension is carried out rigorously and whether the factorization bounds are sharp as stated. Desk rejection would be premature without seeing the details.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to derive factorization results for formal power series over principal ideal domains that yield sharp upper bounds on the number of irreducible factors (up to units), from which several irreducibility criteria follow. It further asserts an extension of the classical Dumas irreducibility criterion to formal power series over discrete valuation domains by invoking the theory of Newton polygons for power series.

Significance. If the central claims hold, the work would supply concrete irreducibility tests for power series rings that are not available from the polynomial case alone, with the sharp bounds constituting a modest but useful strengthening. The explicit use of Newton polygons for the Dumas extension is a natural direction, though its validity hinges on the justification of the polygon construction itself.

major comments (1)

[Abstract] Abstract: the extension of Dumas' criterion is stated to follow from 'theory of Newton polygons for power series' over DVRs, yet the abstract supplies no statement of the required hypotheses (e.g., that v(a_i) → ∞ as i → ∞ so that the lower convex hull is well-defined and finite in each slope segment). This hypothesis is load-bearing for the irreducibility conclusion, as the classical Dumas argument for polynomials relies on finite support.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and the constructive comment on the abstract. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the extension of Dumas' criterion is stated to follow from 'theory of Newton polygons for power series' over DVRs, yet the abstract supplies no statement of the required hypotheses (e.g., that v(a_i) → ∞ as i → ∞ so that the lower convex hull is well-defined and finite in each slope segment). This hypothesis is load-bearing for the irreducibility conclusion, as the classical Dumas argument for polynomials relies on finite support.

Authors: We agree that the abstract would be clearer if it explicitly stated the hypotheses under which the Newton polygon construction applies to formal power series. In the body of the paper the relevant theorems (the extension of Dumas' criterion and the associated irreducibility tests) are formulated for series in the ring where v(a_i) tends to infinity, which guarantees that the lower convex hull is well-defined and consists of finitely many segments. This condition is the natural analogue of finite support in the polynomial setting and is already required for the Newton polygon to be a finite object. We will therefore revise the abstract to include the hypothesis that v(a_i) → ∞ as i → ∞. This change improves readability without affecting the statements or proofs of the main results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rest on classical extensions without self-referential reduction.

full rationale

The paper derives factorization bounds and extends Dumas' criterion to formal power series over DVRs via Newton polygons. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the abstract invokes standard theory of Newton polygons for power series as an established tool rather than deriving it internally. The central claims build on ring-theoretic arguments and classical results with independent content, satisfying the self-contained benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results depend on standard properties of principal ideal domains, discrete valuation domains, and the applicability of Newton polygon techniques to power series; no free parameters or invented entities are described.

axioms (2)

standard math Principal ideal domains and discrete valuation domains satisfy the required ring axioms for factorization and valuation theory
Base setting for all stated results on formal power series.
domain assumption Newton polygon theory extends from polynomials to formal power series
Invoked to extend the Dumas criterion.

pith-pipeline@v0.9.0 · 5607 in / 1155 out tokens · 25351 ms · 2026-05-23T05:43:25.716481+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

?

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

using theory of Newton polygons for power series, we extend the classical Dumas irreducibility criterion to formal power series over discrete valuation domains
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 8 ... Newton polygon ... single edge of negative slope −k/n ... gcd(k,n)=1

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

26 extracted references · 26 canonical work pages

[1]

Dietzel, S

A. Nikseresht and S. Rahimi, Factorization properties of quotien ts of polynomials and power series rings by monomial ideals, Communications in Algebra 52:7 (2024), 2760–2768. https://doi.org/10.1080/00927872.2024.2305686 11

work page doi:10.1080/00927872.2024.2305686 2024
[2]

B. G. Kang, K. A. Loper, T. G. Lucas, M. H. Park, and P. T. Toan , The Krull dimension of power series rings over non-SFT rings, Journal of Pure and Applied Algebra 217 (2013), 254–258. http://dx.doi.org/10.1016/j.jpaa.2012.06.006

work page doi:10.1016/j.jpaa.2012.06.006 2013
[3]

B. G. Kang and M. H. Park, Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable, Journal of Algebra 378 (2013), 12–21. https://doi.org/10.1016/j.jalgebra.2012.05.017

work page doi:10.1016/j.jalgebra.2012.05.017 2013
[4]

Sambale, An invitation to formal power series

B. Sambale, An invitation to formal power series. Jahresbericht Deutschen Mathematiker-Vereinigung 125 (2022), 3–69. https://doi.org/10.1365/s13291-022-00256-6

work page doi:10.1365/s13291-022-00256-6 2022
[5]

Birmajer and J

D. Birmajer and J. B. Gil, Factorization of quadratic polynomials in t he ring of for- mal power series over Z. Journal of Algebra and Its Applications 6:6 (2007), 1027–1037. https://doi.org/10.1142/S021949880700265X

work page doi:10.1142/s021949880700265x 2007
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Birmajer, J

D. Birmajer, J. B. Gil, and M. Weiner, Factoring polynomials in the rin g of for- mal power series over Z. International Journal of Number Theory 8:7 (2012), 1763–1776. https://doi.org/10.1142/S1793042112501011

work page doi:10.1142/s1793042112501011 2012
[7]

Birmajer and J

D. Birmajer and J. B. Gil, Arithmetic in the ring of formal power ser ies with integer coeﬃcients. Amer. Math. Monthly. 115:6 (2008), 541–549. https://doi.org/10.1080/00029890.2008.11920560

work page doi:10.1080/00029890.2008.11920560 2008
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D. W. Hoﬀmann, The Newton polygon of a product of power series . Manuscripta Math. 85 (1994), 109–118. https://doi.org/10.1007/BF02568188

work page doi:10.1007/bf02568188 1994
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Dumas, Sur quelques cas d’irr´ eductibilit´ e des polynomes ´ a co eﬃcients rationnels, J

G. Dumas, Sur quelques cas d’irr´ eductibilit´ e des polynomes ´ a co eﬃcients rationnels, J. Math. Pure Appl. 2 (1906), 191–258. http://eudml.org/doc/233912

work page 1906
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Rond and B

G. Rond and B. Schober, An irreducibility criterion for power ser ies, Proc. Amer. Math. Soc. , 145:11 (2017), 4731–4739. https://doi.org/10.1090/proc/13635

work page doi:10.1090/proc/13635 2017
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Niven, Formal power series

I. Niven, Formal power series. American Mathematical Monthly 76:8 (1969), 871–889. https://doi.org/10.1080/00029890.1969.12000359

work page doi:10.1080/00029890.1969.12000359 1969
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J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. , 177 (1973), 299–304. https://doi.org/10.1090/S0002-9947-1973-0316451-8

work page doi:10.1090/s0002-9947-1973-0316451-8 1973
[14]

J. T. Arnold, Rings with ﬁnite Krull dimension, Indiana University Mathematical Journal , 31 (1982), 897–911. https://doi.org/10.1512/iumj.1982.31.31061

work page doi:10.1512/iumj.1982.31.31061 1982
[15]

Elliot, Factoring formal power series over principal ideal dom ains, Trans

J. Elliot, Factoring formal power series over principal ideal dom ains, Trans. Amer. Math. Soc. , 366:8 (2014), 3997–4019. https://doi.org/10.1090/S0002-9947-2014-05903-5

work page doi:10.1090/s0002-9947-2014-05903-5 2014
[16]

On convergence of stringy motives of wild p^n -cyclic quotient singularities

J. Singh and S. Kumar, A new class of irreducible polynomials, Communications in Algebra , 49:6, (2021), 2722–2727. https://doi.org/10.1080/00927872.2021.1881789

work page doi:10.1080/00927872.2021.1881789 2021
[17]

J. W. Brewer, Power series over commutative rings , Lecture Notes in Pure and Applied Mathematics 64, Marcel Dekker, 1981

work page 1981
[18]

On non-abelian extensions of Leib- niz algebras

Le Thi N. Giau and P. T. Toan, On power series ring over valuation domains, Communications in Algebra 46:5 (2018), 1843–1853. https://doi.org/10.1080/00927872.2017.1346105

work page doi:10.1080/00927872.2017.1346105 2018
[19]

Kang and M

B. Kang and M. H. Park, A localization of a power series ring over a valuation domain, Journal of Pure and Applied Algebra 140 (1999), 107–124. https://doi.org/10.1016/S0022-4049(97)00211-9

work page doi:10.1016/s0022-4049(97)00211-9 1999
[20]

M. H. Park, Root extension of generalized power series rings, Communications in Algebra 52:4 (2024), 1570–1577. https://doi.org/10.1080/00927872.2023.2266850

work page doi:10.1080/00927872.2023.2266850 2024
[21]

M. S. Dutta and H. K. Saikia, Irreducibility of a formal power ser ies with integer coeﬃcients, J. Indian Math. Society (N. S.) 88:(3-4) (2021), 298–308. https://doi.org/10.18311/jims/2021/24897

work page doi:10.18311/jims/2021/24897 2021
[22]

M. S. Dutta and H. K. Saikia, Factorization of a formal power se ries with constant term certain power of a prime, J. Indian Math. Society (N. S.) 91:(3-4) (2024), 344–355. https://doi.org/10.18311/jims/2024/31941

work page doi:10.18311/jims/2024/31941 2024
[23]

Samuel, On unique factorization domains III

P. Samuel, On unique factorization domains III. J. Math. 5 (1961), 1–17. https://doi.org/10.1215/IJM%2F1255629643

work page doi:10.1215/ijm 1961
[24]

Gilmer, A note on the quotient ﬁeld of the domain D[[X]], Proc

R. Gilmer, A note on the quotient ﬁeld of the domain D[[X]], Proc. Amer. Math. Soc. , 18 (1967), 1138–1140. https://doi.org/10.2307/2035816

work page doi:10.2307/2035816 1967
[25]

Gilmer, Multiplicative Ideal Theory , Marcel Dekker, NewYork, (1972)

R. Gilmer, Multiplicative Ideal Theory , Marcel Dekker, NewYork, (1972). 12

work page 1972
[26]

Dietzel, S

S. Hizem, Two absorbing factorization formal power series ring s, Communications in Algebra , 52:11 (2024), 4716–4723. https://doi.org/10.1080/00927872.2024.2357190

work page doi:10.1080/00927872.2024.2357190 2024
[27]

ORCID.png

S. Kumar and J. Singh, A study of some recent irreducibility crite ria for polynomials having integer coeﬃcients. arXiv Preprint: https://arxiv.org/abs/2310.02860 (2023), pp. 1–20. (1,2) Department of Mathematics, Guru Nanak Dev University, Amri tsar-143005, India rishugarg128@gmail.com, jitender.math@gndu.ac.in 13 This figure "ORCID.png" is available in ...

work page arXiv 2023

[1] [1]

Dietzel, S

A. Nikseresht and S. Rahimi, Factorization properties of quotien ts of polynomials and power series rings by monomial ideals, Communications in Algebra 52:7 (2024), 2760–2768. https://doi.org/10.1080/00927872.2024.2305686 11

work page doi:10.1080/00927872.2024.2305686 2024

[2] [2]

B. G. Kang, K. A. Loper, T. G. Lucas, M. H. Park, and P. T. Toan , The Krull dimension of power series rings over non-SFT rings, Journal of Pure and Applied Algebra 217 (2013), 254–258. http://dx.doi.org/10.1016/j.jpaa.2012.06.006

work page doi:10.1016/j.jpaa.2012.06.006 2013

[3] [3]

B. G. Kang and M. H. Park, Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable, Journal of Algebra 378 (2013), 12–21. https://doi.org/10.1016/j.jalgebra.2012.05.017

work page doi:10.1016/j.jalgebra.2012.05.017 2013

[4] [4]

Sambale, An invitation to formal power series

B. Sambale, An invitation to formal power series. Jahresbericht Deutschen Mathematiker-Vereinigung 125 (2022), 3–69. https://doi.org/10.1365/s13291-022-00256-6

work page doi:10.1365/s13291-022-00256-6 2022

[5] [5]

Birmajer and J

D. Birmajer and J. B. Gil, Factorization of quadratic polynomials in t he ring of for- mal power series over Z. Journal of Algebra and Its Applications 6:6 (2007), 1027–1037. https://doi.org/10.1142/S021949880700265X

work page doi:10.1142/s021949880700265x 2007

[6] [6]

Birmajer, J

D. Birmajer, J. B. Gil, and M. Weiner, Factoring polynomials in the rin g of for- mal power series over Z. International Journal of Number Theory 8:7 (2012), 1763–1776. https://doi.org/10.1142/S1793042112501011

work page doi:10.1142/s1793042112501011 2012

[7] [7]

Birmajer and J

D. Birmajer and J. B. Gil, Arithmetic in the ring of formal power ser ies with integer coeﬃcients. Amer. Math. Monthly. 115:6 (2008), 541–549. https://doi.org/10.1080/00029890.2008.11920560

work page doi:10.1080/00029890.2008.11920560 2008

[8] [8]

D. W. Hoﬀmann, The Newton polygon of a product of power series . Manuscripta Math. 85 (1994), 109–118. https://doi.org/10.1007/BF02568188

work page doi:10.1007/bf02568188 1994

[9] [9]

Dumas, Sur quelques cas d’irr´ eductibilit´ e des polynomes ´ a co eﬃcients rationnels, J

G. Dumas, Sur quelques cas d’irr´ eductibilit´ e des polynomes ´ a co eﬃcients rationnels, J. Math. Pure Appl. 2 (1906), 191–258. http://eudml.org/doc/233912

work page 1906

[10] [10]

Rond and B

G. Rond and B. Schober, An irreducibility criterion for power ser ies, Proc. Amer. Math. Soc. , 145:11 (2017), 4731–4739. https://doi.org/10.1090/proc/13635

work page doi:10.1090/proc/13635 2017

[11] [12]

Niven, Formal power series

I. Niven, Formal power series. American Mathematical Monthly 76:8 (1969), 871–889. https://doi.org/10.1080/00029890.1969.12000359

work page doi:10.1080/00029890.1969.12000359 1969

[12] [13]

J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. , 177 (1973), 299–304. https://doi.org/10.1090/S0002-9947-1973-0316451-8

work page doi:10.1090/s0002-9947-1973-0316451-8 1973

[13] [14]

J. T. Arnold, Rings with ﬁnite Krull dimension, Indiana University Mathematical Journal , 31 (1982), 897–911. https://doi.org/10.1512/iumj.1982.31.31061

work page doi:10.1512/iumj.1982.31.31061 1982

[14] [15]

Elliot, Factoring formal power series over principal ideal dom ains, Trans

J. Elliot, Factoring formal power series over principal ideal dom ains, Trans. Amer. Math. Soc. , 366:8 (2014), 3997–4019. https://doi.org/10.1090/S0002-9947-2014-05903-5

work page doi:10.1090/s0002-9947-2014-05903-5 2014

[15] [16]

On convergence of stringy motives of wild p^n -cyclic quotient singularities

J. Singh and S. Kumar, A new class of irreducible polynomials, Communications in Algebra , 49:6, (2021), 2722–2727. https://doi.org/10.1080/00927872.2021.1881789

work page doi:10.1080/00927872.2021.1881789 2021

[16] [17]

J. W. Brewer, Power series over commutative rings , Lecture Notes in Pure and Applied Mathematics 64, Marcel Dekker, 1981

work page 1981

[17] [18]

On non-abelian extensions of Leib- niz algebras

Le Thi N. Giau and P. T. Toan, On power series ring over valuation domains, Communications in Algebra 46:5 (2018), 1843–1853. https://doi.org/10.1080/00927872.2017.1346105

work page doi:10.1080/00927872.2017.1346105 2018

[18] [19]

Kang and M

B. Kang and M. H. Park, A localization of a power series ring over a valuation domain, Journal of Pure and Applied Algebra 140 (1999), 107–124. https://doi.org/10.1016/S0022-4049(97)00211-9

work page doi:10.1016/s0022-4049(97)00211-9 1999

[19] [20]

M. H. Park, Root extension of generalized power series rings, Communications in Algebra 52:4 (2024), 1570–1577. https://doi.org/10.1080/00927872.2023.2266850

work page doi:10.1080/00927872.2023.2266850 2024

[20] [21]

M. S. Dutta and H. K. Saikia, Irreducibility of a formal power ser ies with integer coeﬃcients, J. Indian Math. Society (N. S.) 88:(3-4) (2021), 298–308. https://doi.org/10.18311/jims/2021/24897

work page doi:10.18311/jims/2021/24897 2021

[21] [22]

M. S. Dutta and H. K. Saikia, Factorization of a formal power se ries with constant term certain power of a prime, J. Indian Math. Society (N. S.) 91:(3-4) (2024), 344–355. https://doi.org/10.18311/jims/2024/31941

work page doi:10.18311/jims/2024/31941 2024

[22] [23]

Samuel, On unique factorization domains III

P. Samuel, On unique factorization domains III. J. Math. 5 (1961), 1–17. https://doi.org/10.1215/IJM%2F1255629643

work page doi:10.1215/ijm 1961

[23] [24]

Gilmer, A note on the quotient ﬁeld of the domain D[[X]], Proc

R. Gilmer, A note on the quotient ﬁeld of the domain D[[X]], Proc. Amer. Math. Soc. , 18 (1967), 1138–1140. https://doi.org/10.2307/2035816

work page doi:10.2307/2035816 1967

[24] [25]

Gilmer, Multiplicative Ideal Theory , Marcel Dekker, NewYork, (1972)

R. Gilmer, Multiplicative Ideal Theory , Marcel Dekker, NewYork, (1972). 12

work page 1972

[25] [26]

Dietzel, S

S. Hizem, Two absorbing factorization formal power series ring s, Communications in Algebra , 52:11 (2024), 4716–4723. https://doi.org/10.1080/00927872.2024.2357190

work page doi:10.1080/00927872.2024.2357190 2024

[26] [27]

ORCID.png

S. Kumar and J. Singh, A study of some recent irreducibility crite ria for polynomials having integer coeﬃcients. arXiv Preprint: https://arxiv.org/abs/2310.02860 (2023), pp. 1–20. (1,2) Department of Mathematics, Guru Nanak Dev University, Amri tsar-143005, India rishugarg128@gmail.com, jitender.math@gndu.ac.in 13 This figure "ORCID.png" is available in ...

work page arXiv 2023