## Loose edges and factorization theorems

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- by Janusz Gwoździewicz, Beata Gryszka and Bernd Schober PDF
- Proc. Amer. Math. Soc.
**149**(2021), 2265-2278 Request permission

## Abstract:

Let $R$ be a regular local ring with maximal ideal $\mathfrak {m}$. We consider elements $f \in R$ such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of $f$ to such an edge is a product of two coprime polynomials, then $f$ factorizes in the $\mathfrak {m}$-adic completion.## References

- E. Artal Bartolo, Pi. Cassou-Noguès, I. Luengo, and A. Melle Hernández,
*On $\nu$-quasi-ordinary power series: factorization, Newton trees and resultants*, Topology of algebraic varieties and singularities, Contemp. Math., vol. 538, Amer. Math. Soc., Providence, RI, 2011, pp. 321–343. MR**2777828**, DOI 10.1090/conm/538/10610 - E. Artal Bartolo, I. Luengo, and A. Melle-Hernández,
*High-school algebra of the theory of dicritical divisors: atypical fibers for special pencils and polynomials*, J. Algebra Appl.**14**(2015), no. 9, 1540009, 26. MR**3368261**, DOI 10.1142/S0219498815400095 - Vincent Cossart and Olivier Piltant,
*Resolution of singularities of arithmetical threefolds*, J. Algebra**529**(2019), 268–535. MR**3942183**, DOI 10.1016/j.jalgebra.2019.02.017 - E. R. García Barroso and P. D. González-Pérez,
*Decomposition in bunches of the critical locus of a quasi-ordinary map*, Compos. Math.**141**(2005), no. 2, 461–486. MR**2134276**, DOI 10.1112/S0010437X04001216 - P. D. González Pérez,
*Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant*, Canad. J. Math.**52**(2000), no. 2, 348–368 (French, with French summary). MR**1755782**, DOI 10.4153/CJM-2000-016-8 - Abramo Hefez,
*Irreducible plane curve singularities*, Real and complex singularities, Lecture Notes in Pure and Appl. Math., vol. 232, Dekker, New York, 2003, pp. 1–120. MR**2075059** - Heisuke Hironaka,
*Characteristic polyhedra of singularities*, J. Math. Kyoto Univ.**7**(1967), 251–293. MR**225779**, DOI 10.1215/kjm/1250524227 - Serge Lang,
*Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR**1878556**, DOI 10.1007/978-1-4613-0041-0 - Aleksandar Lipkovski,
*Newton polyhedra and irreducibility*, Math. Z.**199**(1988), no. 1, 119–127. MR**954757**, DOI 10.1007/BF01160215 - Adam Parusiński and Guillaume Rond,
*The Abhyankar-Jung theorem*, J. Algebra**365**(2012), 29–41. MR**2928451**, DOI 10.1016/j.jalgebra.2012.05.003 - Guillaume Rond and Bernd Schober,
*An irreducibility criterion for power series*, Proc. Amer. Math. Soc.**145**(2017), no. 11, 4731–4739. MR**3691990**, DOI 10.1090/proc/13635

## Additional Information

**Janusz Gwoździewicz**- Affiliation: Institute of Mathematics, Pedagogical University of Cracow, Podchora̧\accent95 zych 2, PL-30-084 Cracow, Poland
- Email: janusz.gwozdziewicz@up.krakow.pl
**Beata Gryszka**- Affiliation: Institute of Mathematics, Pedagogical University of Cracow, Podchora̧\accent95 zych 2, PL-30-084 Cracow, Poland
- MR Author ID: 1233841
- Email: bhejmej1f@gmail.com
**Bernd Schober**- Affiliation: Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
- MR Author ID: 1218416
- ORCID: 0000-0003-0315-0656
- Email: schober.math@gmail.com; bernd.schober@uni-oldenburg.de
- Received by editor(s): December 6, 2018
- Received by editor(s) in revised form: April 8, 2019
- Published electronically: March 23, 2021
- Additional Notes: The third author was supported by the DFG-project “Order zeta functions and resolutions of singularities” (DFG project number: 373111162).
- Communicated by: Rachel Pries
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 2265-2278 - MSC (2020): Primary 13F25, 12E05, 14B05
- DOI: https://doi.org/10.1090/proc/14720
- MathSciNet review: 4246781