On irreducible polynomials in several variables which become reducible when the variables are replaced by powers of themselves

Abstract

Let Q(xi, , x,) be a polynomial in xi, , x8, irreducible in the field of all constants. There exist, in certain cases, integers ti, * , t8 such that the polynomial Q(xitl, * * *, xsts) is reducible. The determination of the integers t for which reducibility occurs is a problem which arose in an investigation of J. F. Ritt on the factorization of exponential forms.t The case of real interest is that in which Q has at least three terms. We shall, in this introduction, limit ourselves to the discussion of the results for this case. The relatively simple case of two terms is treated in Part III. For a fairly general, but not perfectly general, type of polynomial Q, consisting of more than two terms, Ritt proved that the sets t break up into a finite number of classes, the sets of any one class being, from a certain point of view, equivalent. In the present paper, we obtain information relative to the sets t which, in certain respects, is final. Our results are embodied in the following theorems, in which each ti is understood to be a positive integer.