# ABSOLUTE IRREDUCIBILITY OF BIVARIATE POLYNOMIALS VIA POLYTOPE METHOD

• Koyuncu, Fatih
• Published : 2011.09.01
• 38 4

#### Abstract

For any field F, a polynomial f $\in$ F[$x_1,x_2,{\ldots},x_k$] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F, i.e., irreducible over every algebraic extension of F. We present some results giving new integrally indecomposable classes of polygons. Consequently, we have some criteria giving many types of absolutely irreducible bivariate polynomials over arbitrary fields.

#### Keywords

absolute irreducibility;bivariate polynomials;integral polygons;integral indecomposability;polytope method

#### References

1. S. Gao, Absolute irreducibility of polynomials via Newton polytopes, J. Algebra 237 (2001), no. 2, 501-520. https://doi.org/10.1006/jabr.2000.8586
2. S. Gao and A. G. B. Lauder, Decomposition of Polytopes and Polynomials, Discrete Comput. Geom. 26 (2001), no. 1, 89-104. https://doi.org/10.1007/s00454-001-0024-0
3. F. Koyuncu and F. Ozbudak, Integral and homothetic indecomposability with applications to irreducibility of polynomials, Turkish J. Math. 33 (2009), no. 3, 283-294.
4. A. Lipkovski, Newton polyhedra and irreducibility, Math. Z. 199 (1998), no. 2, 119-127.
5. A. M. Ostrowski, On multiplication and factorization of polynomials I, Lexicographic orderings and extreme aggregates of terms, Aequationes Math. 13 (1975), no. 3, 201-228. https://doi.org/10.1007/BF01836524
6. A. M. Ostrowski, On multiplication and factorization of polynomials II, Irreducibility discussion, Aequationes Math. 14 (1976), no. 1-2, 1-31. https://doi.org/10.1007/BF01836201
7. G. M. Ziegler, Lectures on Polytopes, GTM 152, Springer-Verlag, 1995.