Factoring formal power series over principal ideal domains

Elliott, Jesse

doi:10.1090/S0002-9947-2014-05903-5

Factoring formal power series over principal ideal domains

Author: Jesse Elliott
Journal: Trans. Amer. Math. Soc. 366 (2014), 3997-4019
MSC (2010): Primary 13F25, 13F10, 13F15, 13A05; Secondary 11S99
DOI: https://doi.org/10.1090/S0002-9947-2014-05903-5
Published electronically: March 26, 2014
MathSciNet review: 3206450
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Abstract: We provide an irreducibility test and factoring algorithm (with some qualifications) for formal power series in the unique factorization domain $R[[X]]$, where $R$ is any principal ideal domain. We also classify all integral domains arising as quotient rings of $R[[X]]$. Our main tool is a generalization of the $p$-adic Weierstrass preparation theorem to the context of complete filtered commutative rings.

References

M. Artin, On the joins of Hensel rings, Advances in Math. 7 (1971), 282–296 (1971). MR 289501, DOI https://doi.org/10.1016/S0001-8708%2871%2980007-5
Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
James W. Brewer, Power series over commutative rings, Lecture Notes in Pure and Applied Mathematics, vol. 64, Marcel Dekker, Inc., New York, 1981. MR 612477
Daniel Birmajer and Juan B. Gil, Arithmetic in the ring of formal power series with integer coefficients, Amer. Math. Monthly 115 (2008), no. 6, 541–549. MR 2416254, DOI https://doi.org/10.1080/00029890.2008.11920560
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Factorization of quadratic polynomials in the ring of formal power series over $\Bbb Z$, J. Algebra Appl. 6 (2007), no. 6, 1027–1037. MR 2376798, DOI https://doi.org/10.1142/S021949880700265X
Daniel Birmajer, Juan B. Gil, and Michael Weiner, Factoring polynomials in the ring of formal power series over $\Bbb Z$, Int. J. Number Theory 8 (2012), no. 7, 1763–1776. MR 2968949, DOI https://doi.org/10.1142/S1793042112501011
David G. Cantor and Daniel M. Gordon, Factoring polynomials over $p$-adic fields, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 185–208. MR 1850606, DOI https://doi.org/10.1007/10722028_10
Alexandre L. Chistov, Efficient factoring polynomials over local fields and its applications, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 1509–1519. MR 1159333
Robert Gilmer, Some questions for further research, Multiplicative ideal theory in commutative algebra, Springer, New York, 2006, pp. 405–415. MR 2265822, DOI https://doi.org/10.1007/978-0-387-36717-0_24
Jordi Guàrdia, Enric Nart, and Sebastian Pauli, Single-factor lifting and factorization of polynomials over local fields, J. Symbolic Comput. 47 (2012), no. 11, 1318–1346. MR 2927133, DOI https://doi.org/10.1016/j.jsc.2012.03.001
Jordi Guàrdia, Jesús Montes, and Enric Nart, Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields, J. Théor. Nombres Bordeaux 23 (2011), no. 3, 667–696 (English, with English and French summaries). MR 2861080, DOI https://doi.org/10.5802/jtnb.782
Jordi Guàrdia, Jesús Montes, and Enric Nart, Okutsu invariants and Newton polygons, Acta Arith. 145 (2010), no. 1, 83–108. MR 2719575, DOI https://doi.org/10.4064/aa145-1-5
Serge Lang, Algebra, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. MR 783636
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461

James M. McDonough, Integral domains arising as quotient rings of $\mathbb {Z}[[x]]$, Master’s Thesis, California State University, Channel Islands, 2011.

Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856

Matthew O’Malley, On the Weierstrass preparation theorem, Rocky Mountain J. Math. 2 (1972), no. 2, 265–273. MR 289500, DOI https://doi.org/10.1216/RMJ-1972-2-2-265

Sebastian Pauli, Factoring polynomials over local fields, J. Symbolic Comput. 32 (2001), no. 5, 533–547. MR 1858009, DOI https://doi.org/10.1006/jsco.2001.0493

Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237

Hiroki Sumida, Greenberg’s conjecture and the Iwasawa polynomial, J. Math. Soc. Japan 49 (1997), no. 4, 689–711. MR 1466360, DOI https://doi.org/10.2969/jmsj/04940689