Nearly irreducibility of polynomials and the Newton diagrams

Mateusz Masternak

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


irreducibility of polynomials; Newton diagram; Newton polygon; plane algebraic curve

Mathematics Subject Classification (2010)


11R09; 52B20; 14H50

References


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.

Ajzenberg L.A., and A.P. Južakow. Integral representations and residues in multidimentional complex analysis. , Novosibirsk: Izdat. Nauka, Sibirskoe Otdelenie, 1991. (in Russian).

Atiyah, Michael F. "Angular momentum, convex polyhedra and algebraic geometry." Proc. Edinburgh Math. Soc. (2) 26, no. 2, (1983): 121-133.

Bernstein, D.N. "The number of roots of a system of equations." Funkcional. Anal. i Priložen. 9, no. 3 (1975): 1-4. (in Russian)

Bernstein, D.N., A.G. Kušnirenko, and A.G. Hovanski˘ı. "Newton polyhedra." Uspehi Mat. Nauk 31, no. 3(189) (1976): 201-202. (in Russian)

Cassou-Noguè s, Pierrette, and Arkadiusz Płoski. "Invariants of plane curve singularities and Newton diagrams." Univ. Iagel. Acta Math. 49 (2011): 9-34.

Fulton, William. Algebraic curves. An introduction to algebraic geometry. New York-Amsterdam: W.A. Benjamin, Inc., 1969.

Hovanskiı, A.G. "Newton polyhedra, and toroidal varieties." Funkcional. Anal. i Priložen. 11, no. 4 (1977) 56-64. (in Russian)

Hovanskiı, A.G. "Newton polyhedra, and the genus of complete intersections." Funktsional. Anal. i Prilozhen. 12, no. 1, (1978): 51-61. (in Russian)

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).

Kušnirenko, A.G. "Newton polyhedra and Bezout’s theorem." Funkcional. Anal. i Priložen. 10, no. 3 (1976): 82-83. (in Russian).

Kušnirenko, A.G. "Polyèdres de Newton et nombres de Milnor." Invent. Math. 32, no. 1 (1976): 1-31.

Masternak, Mateusz. "Invariants of singularities of polynomials in two complex variables and the Newton diagrams." Univ. Iagel. Acta Math. 39 (2001): 179-188.

Ostrowski, A.M. "Uber die Bedeutung der Theorie der konvexen Polyeder fur die formale Algebra." Jahresberichte Deutsche Math. Verein 30 (1921): 98-99.

Płoski, Arkadiusz. "Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of C2." Ann. Polon. Math. 51 (1990): 275-281.

Płoski, Arkadiusz. "On the irreducibility of polynomials in several complex variables." Bull. Polish Acad. Sci. Math. 39, no. 3-4 (1991): 241-247.

Rubel, L. A., A. Schinzel, and H. Tverberg. "On difference polynomials and hereditarily irreducible polynomials." J. Number Theory 12, no. 2 (1980): 230-235.

Schneider, Rolf. Convex bodies: the Brunn-Minkowski theory. Vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 1993.

Webster, Roger. Convexity. New York: The Clarendon Press, Oxford University Press, 1994.


Full Text: PDF

Download statistics: 552

Licencja Creative Commons
This article by Mateusz Masternak is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej