Complexity Classes and Completeness in Algebraic Geometry

We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting. Valiant's theory of algebraic/arithmetic complexity classes is an algebraic analogue of Boolean complexity theory, where Boolean functions are replaced by polynomials over any ring, and Boolean circuits are replaced by arithmetic circuits that use the ×, + operations of the ring instead of the Boolean operations. An interesting point about Valiant's theory is how sequences in VNP are made. Keeping with the Boolean procedure of taking an efficiently computable 'verifier' g n