Undecidability of parametric solutions of polynomial equations

Abstract

We prove that, for any field F {\mathbf {F}} of characteristic 0 0 satisfying a hypothesis related to not being algebraically closed, the problem of finding non-constant parametric solutions in F ( t ) {\mathbf {F}}(t) to a polynomial system with coefficients in F {\mathbf {F}} is algorithmically unsolvable.