Dedekind Zeta Zeroes and Faster Complex Dimension Computation∗ J. Maurice Rojas† Yuyu Zhu‡ Texas A&M University Texas A&M University College Station, Texas College Station, Texas
[email protected] [email protected] S k ABSTRACT F 2 (Z[x1;:::; xn]) k;n 2Z Thanks to earlier work of Koiran, it is known that the truth of the has a complex root. While the implication FEASC 2 P =) P=NP has Generalized Riemann Hypothesis (GRH) implies that the dimension long been known, the inverse implication FEASC < P =) P , NP of algebraic sets over the complex numbers can be determined remains unknown. Proving the implication FEASC < P =) P,NP within the polynomial-hierarchy. The truth of GRH thus provides a would shed new light on the P vs. NP Problem, and may be easier direct connection between a concrete algebraic geometry problem than attempting to prove the complexity lower bound FEASC < P and the P vs. NP Problem, in a radically different direction from (whose truth is still unknown). the geometric complexity theory approach to VP vs. VNP. We Detecting complex roots is the D =0 case of the following more explore more plausible hypotheses yielding the same speed-up. general problem: One minimalist hypothesis we derive involves improving the error [ DIM ; 2 N × Z ;:::; k term (as a function of the degree, coefficient height, and x) on the C: Given (D F ) ( [x1 xn]) , fraction of primes p ≤ x for which a polynomial has roots mod p. k;n 2Z decide whether the complex zero set of F has A second minimalist hypothesis involves sharpening current zero- dimension at least D.