Loose edges and factorization theorems
HTML articles powered by AMS MathViewer
- by Janusz Gwoździewicz, Beata Gryszka and Bernd Schober
- Proc. Amer. Math. Soc. 149 (2021), 2265-2278
- DOI: https://doi.org/10.1090/proc/14720
- Published electronically: March 23, 2021
- PDF | 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
Bibliographic 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