Cook's versus Valiant's hypothesis (Q1978701)

Valiant developed a nonuniform algebraic analogue of the theory of NP-completeness for computations with polynomials over a field \(k\) in [\textit{L. G. Valiant}, Proceedings of the 11th ACM STOC, 1979, pp. 249-261; Logic and algorithmic, int. Symp., Zürich 1980, Monogr. L'Enseign. Math. 30, 365-380 (1982; Zbl 0474.68062)]. This theory centers around his hypothesis VP\(_{k}\neq\)VNP\(_{k}\), the analogue of Cook's hypothesis P\(\neq\)NP. We identify the Boolean parts of Valiant's algebraic complexity classes VP\(_{k}\) and VNP\(_{k}\) as familiar nonuniform complexity classes. As a consequence, we obtain rather strong evidence for Valiant's hypothesis: if it were wrong, then the nonuniform versions of NC and PH would be equal. In particular, the polynomial hierarchy would collapse to the second level. We show this for fields of characteristic zero and finite fields; in the first case we assume a generalized Riemann hypothesis. The crucial step in our proof is the elimination of constants in \(k\), which relies on a recent method proposed by \textit{P. Koiran} [J. Complexity 12, No. 4, 273-286, Art. No. 0019 (1996; Zbl 0862.68053)].