Types of singularity of components of difference polynomials

Abstract

Let S be a set of difference polynomials. The perfect difference ideal {S} [2, p. 76 and p. 82] may properly contain the difference ideal ~/[S]. It follows that in determining the irreducible components of the manifold of S it is not sufficient to consider only factorizations of polynomials of n/IS ] (or, equivalently, of IS]). In particular, let S contain only the algebraically irreducible difference polynomial F, and let M be a singular component [2, p. 161] of the manifold ofF. M will be called strongly singular if it can be found from factorizations in [F], and weakly singular otherwise. This definition will be made precise in Section 1. By analogy, all singular components of a differential polynomial with coefficients in a field of characteristic 0 should be considered strongly singular, since if G is such a polynomial, ~/[G] = {G}. An examination of the previously known examples of singular components of difference polynomials reveals that they are all strongly singular. This is not surprising in itself, since it appears to be difficult to prove that a manifold is a component if it is weakly singular. However, the examples cover three rather large classes of singular components, namely, those described in Theorems I, II, and III of Section 1, as well as one example with markedly different properties [2, p. 332]. It would indicate a most unexpected simplicity in the structure of the perfect ideals generated by algebraically irreducible difference polynomials if the known examples were an indication of the general situation. The principal object of this paper is to remove this possibility by providing an example (Section 2) of a difference polynomial F with the weakly singular component M = {0}. This example is of order 5, and Theorem I shows that it could not be of order less than 3. Whether examples of orders 3 or 4 exist is not known. In Section 3 approximations to solutions in singular components are considered for the case of coefficients and solutions which are complex valued functions on the integers. Using the polynomial Q of the example of Section 2 it is shown that the solution 0 constituting the singular component can be-approximated uniformly by solutions which are nowhere 0. This is quite unexpected. (Compare [3, p. 259].) It is easily seen to be possible only because of two circumstances: 1) {0} is a weakly singular component, 2) the approximating solutions are not regular, that is they do not lie in difference rings which are integral domains. Although no general theorem can be stated at present, this example suggests an important distinction in the behavior of weakly and strongly singular components, The distinction between weak and strong singularity is only one of an infinite sequence of similar distinctions that may be made using, in place of the difference ideal [F] generated by the difference polynomial F, each ideal of the sequence whose union is {F}, as described in [2, Ch. 3, Section 2].