A complex analogue of Toda's Theorem

TL;DR

This paper extends Toda's theorem to the complex projective setting using topological methods, demonstrating the Poincaré polynomial's key role in computational complexity over complex and real numbers.

Contribution

It develops a complex analogue of Toda's theorem by employing the complex join of quasi-projective varieties, expanding the topological approach to the complex case.

Findings

Established a complex version of Toda's theorem.

Showed the Poincaré polynomial enables polynomial-time decision of the polynomial hierarchy.

Extended topological techniques from real to complex algebraic geometry.

Abstract

Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class $\mathbf{P}^{#\mathbf{P}}$, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class $#\mathbf{P}$. This result, which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines \cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in \cite{BZ09} is topological in nature in which the properties of the topological join is used in a fundamental way. However, the constructions used in \cite{BZ09} were semi-algebraic -- they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in \cite{BZ09} to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence we obtain a complex analogue of Toda's theorem. The results contained in this paper, taken together with those contained in \cite{BZ09}, illustrate the central role of the Poincar\'e polynomial in algorithmic algebraic geometry, as well as, in computational complexity theory over the complex and real numbers -- namely, the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.