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Being motivated by some methods for construction of homothetically indecomposable polytopes, we obtain new methods for construction of families of integrally indecomposable polytopes. As a result, we find new infinite families of absolutely irreducible multivariate polynomials over any field F. Moreover, we provide different proofs of some of the main… (More)
Motivated by the Dubickas's result in , which computes the probability of the irreducible polynomials by Eisenstein's criterion for some families of polynomials in [x], we calculate the probabilities which represent the ratio of absolutely irreducible multivariate polynomials by the polytope method in some families of polynomials over arbitrary fields.
For any field F, a polynomial f ∈ F [x 1 , x 2 ,. .. , 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… (More)
In , Kaltofen introduced a geometric open problem about mul-tivariate polynomials in the sense of exponents of the terms. In this study, by using Newton polytope method, we have provided a solution of this problem for some special cases.