Diophantine equations arising from cubic number fields

Diophantine Equations in Cyclotomic Fields

where p is a given rational prime? It is almost trivial (from the theory of the Gaussian sum or otherwise) that a solution exists with g =p; it is less trivial that a solution also exists when g = p+p+l; but it is not asserted that solutions do not exist for other values of g. While we are unable to give anything like a complete answer to the problem proposed, we can prove something in this dir...

Class-number Problems for Cubic Number Fields

Diophantine Equations and Congruences over Function Fields

We generalize the methods of Pierce of counting solutions to the congruence X ≡ Y b mod D [8] and the square sieve method of counting squares in the sequence f(X) + g(Y ) [7] to the function field setting.

Diophantine Sets of Polynomials over Number Fields

Let R be a number field or a recursive subring of a number field and consider the polynomial ring R[T ]. We show that the set of polynomials with integer coefficients is diophantine over R[T ]. Applying a result by Denef, this implies that every recursively enumerable subset of R[T ]k is diophantine over R[T ].

The Number of Solutions of Diophantine Equations

0. Introduction. In two recent papers 4], 30], Erdd os, Stewart and the author showed that certain diophantine equations have many solutions. In this way they indicated how far certain results are capable for improvements at most. First we mention some relevant results from the literature on upper bounds for the numbers of solutions of diophantine equations and then we sketch how our method lea...