We prove that, for any field F of characteristic 0 satisfying a hypothesis related to not being algebraically closed, the problem of finding nonconstant parametric solutions in F(r) to a polynomial system with coefficients in F is algorithmically unsolvable. Solutions of Diophantine equations over rings and fields are often expressible in terms of polynomials or rational functions. That is, given an equation system over a ring 3? we find nonconstant elements of 3'[t] that satisfy the equation.

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... ote that this is not obviously equivalent to the Diophantine problem in 3t [t] since, in the latter case, we can have coefficients in 3? [t]. Two reasons for interest in the parametric problem are that it relates to the case of Diophantine equations over Q since many systems over Q have parametric solutions and that it corresponds to nonconstant maps of affine varieties. After the Matijasevitch-Davis-Putnam-Robinson proof that the Diophantine problem is unsolvable over Z, Denef and coworkers extended this result to various rings of algebraic integers and to 32[t] for 32 of characteristic zero (but did not treat the parametric problem) [DI]. In [D2] he dealt with 3?[f] for 31 of positive characteristic. See [C, BDL] for related work on these questions. Recently, Pheidas, in unpublished work using methods of [P], proved Diophantine undecidability of 3?[t] for 3?, a finite field, and we dealt with Q(tx, t2), see [KR]. Pheidas also more recently proved unsolvability of a parametric problem for general rings 3Z[t] but his problem is not the same as ours, in that he proves undecidability of nonconstancy in a chosen variable and thus does not establish undecidability of nonconstant maps of affine varieties. Neither his proof nor ours extends to settle our parametric problem for C[t], and we feel this probably lies deeper within algebraic geometry. Definition. The polynomial parametric problem for the given ring 31 is given a finite system of polynomial equations over 3t /t-i(xx, ... , x") = 0