  We define a theory of parameterized algebraic complexity classes in analogy
to parameterized Boolean counting classes. We define the classes VFPT and
VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define
appropriate reductions and completeness notions. Our main contribution is the
VW[1]-completeness proof of the parameterized clique family. This proof is far
more complicated than in the Boolean world. It requires some new concepts like
composition theorems for bounded exponential sums and Boolean-arithmetic
formulas. In addition, we also look at two polynomials linked to the permanent
with vastly different parameterized complexity.
