Nonstandard arithmetic of Hilbert subsets
For an irreducible polynomial f(X,Y) in R[X,Y], J(f) denotes the set of those r in R for which f(r,Y) is reducible. Thus, in the case of \(R={\mathbb{Z}}\), J(f) is the complement of the Hilbert subset of \({\mathbb{Z}}\) corresponding to f. Hilbert's irreducibility theorem says that all Hilbert subsets of \({\mathbb{Z}}\) are infinite. The paper discusses the problem of the size of J(f). In the case of \(R={\mathbb{Z}}\), a sufficient condition for finiteness of J(f) is given, the condition also yields an upper bound for J(f) in terms of the coefficients and the degree of f. In the case of \(R=K[t]\) and an irreducible f in K[X,Y,t], where K is a field and t is transcendental over K, an analogous condition is used to obtain an upper bound for the degree of those \(\phi\) (t) in K[t] for which f(\(\phi\) (t),Y,t) is reducible. Both results are proved by Robinson's methods of nonstandard arithmetic.
scientific article; zbMATH DE number 4212039