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
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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 Neciporuk’s [Soviet Math. Doklady, 7 (1966), pp. 999–1000] 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 Neciporuk’s 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. 1537–1540]. We consider the fundamental problem of solving quadratic systems of equations in @math variables, where @math , @math and @math is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data @math and @math as soon as the ratio @math between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have @math and prove that our algorithms achieve a statistical accuracy, which is nearly un-improvable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size---hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size. Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ( @math ) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex @math of dimension greater than @math , the boundary matrix @math is totally unimodular if and only if @math is torsion-free for all pure subcomplexes @math in @math of dimensions @math and @math , respectively, where @math . Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under @math coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal @math -cycle in a given homology class for any triangulation of an orientable compact @math -manifold or for any finite simplicial complex embedded in @math . Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity. Given a static array of @math totally ordered objects, the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on-line queries of the form “what is the position of a minimum element in the subarray ranging from @math to @math ?”. We focus on two settings, where (1) the input array is available at query time, and (2) the input array is available only at construction time. In setting (1), we show new data structures (a) of size @math bits and query time @math for any positive integer function @math for an arbitrary constant @math , or (b) with @math bits and @math query time, where @math denotes the empirical entropy of @math th order of the input array. In setting (2), we give a data structure of size @math bits and query time @math . All data structures can be constructed in linear time and almost in-place.