2,522 research outputs found
Diophantine equations in two variables
This is an expository lecture on the subject of the title delivered at the
Park-IAS mathematical institute in Princeton (July, 2000).Comment: Not for separate publicatio
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
Convex Combinatorial Optimization
We introduce the convex combinatorial optimization problem, a far reaching
generalization of the standard linear combinatorial optimization problem. We
show that it is strongly polynomial time solvable over any edge-guaranteed
family, and discuss several applications
Laminated Wave Turbulence: Generic Algorithms III
Model of laminated wave turbulence allows to study statistical and discrete
layers of turbulence in the frame of the same model. Statistical layer is
described by Zakharov-Kolmogorov energy spectra in the case of irrational
enough dispersion function. Discrete layer is covered by some system(s) of
Diophantine equations while their form is determined by wave dispersion
function. This presents a very special computational challenge - to solve
Diophantine equations in many variables, usually 6 to 8, in high degrees, say
16, in integers of order and more. Generic algorithms for solving
this problem in the case of
{\it irrational} dispersion function have been presented in our previous
papers. In this paper we present a new generic algorithm for the case of {\it
rational} dispersion functions. Special importance of this case is due to the
fact that in wave systems with rational dispersion the statistical layer does
not exist and the general energy transport is governed by the discrete layer
alone.Comment: submitted to IJMP
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
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