[FOM] Is this known to be undecidable?

[joeshipman@aol.com]

Given: a multivariable polynomial p(x1,x2,...,x_n,y) and a univariate 
polynomial q(z), in the language of fields.

Assume the axioms for fields, and the axioms for characteristic 0 if 
necessary.

Is it decidable whether

Forall (x1,...,xn) Thereexists y p(x1,...,x_n,y)=0

implies

Thereexists z q(z)=0

?

In other words, if I can solve all equations of type p, can I solve 
equation q?

(If it is undecidable when I generalize to allow more existentially 
quantified variables in p and q, that would also be worth knowing, but 
I am mainly interested in the specific case of only one existentially 
quantified variable, which seems quite difficult enough!)

-- JS

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