On the irreducibility of a class of polynomials, IV

K. Győry

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We answer three reducibility (or irreducibility) questions for $0, 1$ -polynomials, those polynomials which have every coefficient either $0$ or $1$ . The first concerns whether a naturally occurring sequence of reducible polynomials is finite. The second is whether every nonempty finite subset of an infinite set of positive integers can be the set of positive exponents of a reducible $0, 1$ -polynomial. The third is the analogous question for exponents of irreducible $0, 1$ -polynomials.

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1. Introduction. The purpose of this paper is to establish some general finiteness results (cf. Theorems 1 and 2) for resultant equations over an arbitrary finitely generated integral domain R over ℤ. Our Theorems 1 and 2 improve and generalize some results of Wirsing [25], Fujiwara [6], Schmidt [21] and Schlickewei [17] concerning resultant equations over ℤ. Theorems 1 and 2 are consequences of a finiteness result (cf. Theorem 3) on decomposable form equations over R. Some applications...