Dedekind Zeta Zeroes and Faster Complex Dimension Computation

TL;DR

This paper investigates alternative hypotheses to GRH, involving Dedekind zeta zeroes and prime distribution errors, which could enable complex algebraic set dimension computation within the polynomial hierarchy.

Contribution

It proposes new, plausible hypotheses that relax GRH assumptions, allowing algebraic geometry problems to be solved efficiently within the polynomial hierarchy.

Findings

Hypotheses involving improved error bounds on prime distributions for polynomials.

Refined zero-free regions for Dedekind zeta functions.

Potential polynomial hierarchy algorithms under these hypotheses.

Abstract

Thanks to earlier work of Koiran, it is known that the truth of the Generalized Riemann Hypothesis (GRH) implies that the dimension of algebraic sets over the complex numbers can be determined within the polynomial-hierarchy. The truth of GRH thus provides a direct connection between a concrete algebraic geometry problem and the P vs.NP Problem, in a radically different direction from the geometric complexity theory approach to VP vs. VNP. We explore more plausible hypotheses yielding the same speed-up. One minimalist hypothesis we derive involves improving the error term (as a function of the degree, coefficient height, and $x$) on the fraction of primes $p\!\leq\!x$ for which a univariate polynomial has roots mod $p$. A second minimalist hypothesis involves sharpening current zero-free regions for Dedekind zeta functions. Both our hypotheses allow failures of GRH but still enable complex dimension computation in the polynomial hierarchy.