1. bookVolume 19 (2020): Issue 1 (December 2020)
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11 Dec 2014
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access type Open Access

Nearly irreducibility of polynomials and the Newton diagrams

Published Online: 31 Dec 2020
Page range: 65 - 77
Received: 07 May 2019
Accepted: 15 Nov 2019
Journal Details
License
Format
Journal
First Published
11 Dec 2014
Publication timeframe
1 time per year
Languages
English
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.

Keywords

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[19] Webster, Roger. Convexity. New York: The Clarendon Press, Oxford University Press, 1994. Cited on 71.Search in Google Scholar

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