Nearly irreducibility of polynomials and the Newton diagrams

Mateusz Masternak

doi:10.2478/aupcsm-2020-0006

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Nearly irreducibility of polynomials and the Newton diagrams

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

Volume 19 (2020): Issue 1 (December 2020)

By: Mateusz Masternak

Open Access

|Dec 2020

Abstract

Let f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero in C². In the paper we give a criterion of nearly irreducibility for a given polynomial f in terms of its Newton diagram.

References

[1] Abhyankar, Shreeram S., and Lee A. Rubel. “Every difference polynomial has a connected zero-set.” J. Indian Math. Soc. (N.S.) 43, no. 1-4 (1979): 69-78. Cited on 66.
Search in Google Scholar Back to article
[2] Ajzenberg L.A., and A.P. Južakow. Integral representations and residues in multidimentional complex analysis., Novosibirsk: Izdat. Nauka, Sibirskoe Otdelenie, 1991. (in Russian). Cited on 70.
Search in Google Scholar Back to article
[3] Atiyah, Michael F. “Angular momentum, convex polyhedra and algebraic geometry.” Proc. Edinburgh Math. Soc. (2) 26, no. 2, (1983): 121-133. Cited on 68.
Search in Google Scholar Back to article
[4] Bernstein, D.N. “The number of roots of a system of equations.” Funkcional. Anal. i Priložen. 9, no. 3 (1975): 1-4. (in Russian) Cited on 68 and 70.
Search in Google Scholar Back to article
[5] Bernstein, D.N., A.G. Kušnirenko, and A.G. Hovanskiĭ. “Newton polyhedra.” Uspehi Mat. Nauk 31, no. 3(189) (1976): 201-202. (in Russian) Cited on 68.
Search in Google Scholar Back to article
[6] Cassou-Noguè s, Pierrette, and Arkadiusz Płoski. “Invariants of plane curve singularities and Newton diagrams.” Univ. Iagel. Acta Math. 49 (2011): 9-34. Cited on 70.
Search in Google Scholar Back to article
[7] Fulton, William. Algebraic curves. An introduction to algebraic geometry. New York-Amsterdam: W.A. Benjamin, Inc., 1969. Cited on 69.
Search in Google Scholar Back to article
[8] Hovanskiĭ, A.G. “Newton polyhedra, and toroidal varieties.” Funkcional. Anal. i Priložen. 11, no. 4 (1977) 56-64. (in Russian) Cited on 68.
Search in Google Scholar Back to article
[9] Hovanskiĭ, A.G. “Newton polyhedra, and the genus of complete intersections.” Funktsional. Anal. i Prilozhen. 12, no. 1, (1978): 51-61. (in Russian) Cited on 68.
Search in Google Scholar Back to article
[10] Kušnirenko, A.G. “Newton polyhedra and a number of roots of a system of k equations in k variables.” Uspehi Mat. Nauk 30, no. 2 (1975): 266-267 (in Russian). Cited on 68, 70 and 74.
Search in Google Scholar Back to article
[11] Kušnirenko, A.G. “Newton polyhedra and Bezout’s theorem.” Funkcional. Anal. i Priložen. 10, no. 3 (1976): 82-83. (in Russian). Cited on 68.
Search in Google Scholar Back to article
[12] Kušnirenko, A.G. “Polyèdres de Newton et nombres de Milnor.” Invent. Math. 32, no. 1 (1976): 1-31. Cited on 68.
Search in Google Scholar Back to article
[13] Masternak, Mateusz. “Invariants of singularities of polynomials in two complex variables and the Newton diagrams.” Univ. Iagel. Acta Math. 39 (2001): 179-188. Cited on 70 and 74.
Search in Google Scholar Back to article
[14] Ostrowski, A.M. “Über die Bedeutung der Theorie der konvexen Polyeder für die formale Algebra.” Jahresberichte Deutsche Math. Verein 30 (1921): 98-99. Cited on 69.
Search in Google Scholar Back to article
[15] Płoski, Arkadiusz. “Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of C².” Ann. Polon. Math. 51 (1990): 275-281. Cited on 70 and 74.
Search in Google Scholar Back to article
[16] Płoski, Arkadiusz. “On the irreducibility of polynomials in several complex variables.” Bull. Polish Acad. Sci. Math. 39, no. 3-4 (1991): 241-247. Cited on 66.
Search in Google Scholar Back to article
[17] Rubel, L. A., A. Schinzel, and H. Tverberg. “On difference polynomials and hereditarily irreducible polynomials.” J. Number Theory 12, no. 2 (1980): 230-235. Cited on 66.
Search in Google Scholar Back to article
[18] Schneider, Rolf. Convex bodies: the Brunn-Minkowski theory. Vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 1993. Cited on 69 and 70.
Search in Google Scholar Back to article
[19] Webster, Roger. Convexity. New York: The Clarendon Press, Oxford University Press, 1994. Cited on 71.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/aupcsm-2020-0006 | Journal eISSN: 2300-133X | Journal ISSN: 2081-545X

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Language: English

Page range: 65 - 77

Submitted on: May 7, 2019

Accepted on: Nov 15, 2019

Published on: Dec 31, 2020

Published by: Pedagogical University of Cracow

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

rreducibility of polynomials,

Newton diagram,

Newton polygon,

plane algebraic curve

Related subjects:

Mathematics,

General mathematics

© 2020 Mateusz Masternak, published by Pedagogical University of Cracow
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 License.

Volume 19 (2020): Issue 1 (December 2020)