Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN in P implies P=NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication that HN not in NP implies P is not equal to NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.
This result immediately implies that @math can be solved by solving a linear system with @math variables and equations over the rationals (with total bit-size @math ), thus easily yielding @math . That @math then follows immediately from the fact that linear algebra can be efficiently parallelized @cite_7 .
@cite_7
2113097540
We establish a lower bound for the formula size of quolynomials over arbitrary fields. Our basic formula operations are addition, subtraction, multiplication and division. The proof is based on Neciporuks [Soviet Math. Doklady, 7 (1966), pp. 9991000] lower bound for Boolean functions and uses formal power series. This result immediately yields a lower bound for the formula size of rational functions over infinite fields. We also show how to adapt Neciporuks method to rational functions over finite fields. These results are then used to show that, over any field, the @math determinant function has formula size at least @math . We thus have an algebraic analogue to the @math lower bound for the Boolean determinant due to Kloss [Soviet Math. Doklady, 7 (1966), pp. 15371540].