The Diophantine Problem for Polynomial Rings and Fields of Rational Functions

Abstract

We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.1. Introduction.3During the last thirty years much work has been done to prove that the elementary theory of various rings is undecidable; see J. Ax [1], Yu.Ersov [9], [10], A. Malcev [14], Yu.Penzin [15], J. Robinson [17]-[20], R. M. Robinson [21], [22] and A. Tarski [23].After M. Davis, Yu.Matijasevic, H. Putnam and J. Robinson (see, e.g., [4], [6]) proved that the existential theory of Z is undecidable, it is natural to ask whether the existential theory of various other rings is undecidable too.Let R be a commutative ring with unity and let R' be a subring of R. We say that the diophantine problem for R with coefficients in R' is unsolvable (solvable) if there exists no (an) algorithm to decide whether or not a polynomial equation (in several variables) with coefficients in R' has a solution in R.In [7] we proved that the diophantine problem for the ring of algebraic integers in any quadratic extension of Q is unsolvable, and recently we have extended this to some more algebraic integer rings.For some very interesting related results, see L. Lipshitz [13].The main results of this paper are:Theorem A. Let R be an integral domain of characteristic zero; then the diophantine problem for R[T] with coefficients in Z[T] is unsolvable.(R[T] denotes the ring of polynomials over R, in one variable T.) Theorem B. Let K be a formally real field, i.e. -1 not the sum of squares