A census of zeta functions of quartic K3 surfaces over F_2

We compute the complete set of candidates for the zeta function of a K3 surface over F_2 consistent with the Weil conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over F_2. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda-Tate theorem for transcendental zeta functions of K3 surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.Comment: 11 pages; final version, minor changes; to appear in ANTS XI

Root finding with threshold circuits

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur

Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation

We propose a new simple but nearly optimal algorithm for the approximation of all sufficiently well isolated complex roots and root clusters of a univariate polynomial. Quite typically the known root-finders at first compute some crude but reasonably good approximations to well-conditioned roots (that is, those isolated from the other roots) and then refine the approximations very fast, by using Boolean time which is nearly optimal, up to a polylogarithmic factor. By combining and extending some old root-finding techniques, the geometry of the complex plane, and randomized parametrization, we accelerate the initial stage of obtaining crude to all well-conditioned simple and multiple roots as well as isolated root clusters. Our algorithm performs this stage at a Boolean cost dominated by the nearly optimal cost of subsequent refinement of these approximations, which we can perform concurrently, with minimum processor communication and synchronization. Our techniques are quite simple and elementary; their power and application range may increase in their combination with the known efficient root-finding methods.Comment: 12 pages, 1 figur

Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a rational solution to f_1=...=f_m=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PP^{NP^NP}, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems.Comment: Slight revision of final journal version of an extended abstract which appeared in STOC 1999. This version includes significant corrections and improvements to various asymptotic bounds. Needs cjour.cls to compil