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Computational Arithmetic Geometry I Sentences Nearly in the Polynomial Hierarchy.pdf

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Computational Arithmetic Geometry I Sentences Nearly in the Polynomial Hierarchy

Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy1 J. Maurice Rojas2 E-mail: rojas@ Department of Mathematics, Texas AM University, College Station, Texas 77843-3368, USA. DEDICATED TO GRETCHEN DAVIS. We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I Given a polynomial f ∈Z[v, x, y], decide the sentence ?v ?x ?y f(v, x, y) ? =0, with all three quantifiers ranging over N (or Z). II Given polynomials f1, . . . , fm∈Z[x1, . . . , xn] with m≥n, decide if there is a rational solution to f1= · · · =fm = 0. We show that problem (I) can be done within coNP for almost all inputs. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that problem (II) can be solved within the complexity class PNP NP for almost all inputs, 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 prac- tical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems. 1. INTRODUCTION AND MAIN RESULTS The negative solution of Hilbert’s Tenth Problem [Mat70, Mat93] has all but dashed earlier hopes of solving large polynomial systems over the integers. However, an immediate positive consequence is the creation of a rich and diverse garden of 1From Journal of Computer and System Sciences, vol. 62, no. 2, march 2001, pp. 216–235. Please see “Notes Added in Proof” near the end of the paper for any changes from the published version. This research was supported by Hong Kong UGC Grant #9040469-730. 2URL: /~rojas . 1 2 J. MAURICE ROJAS hard problems