VPSPACE and a transfer theorem over the complex field

A dichotomy theorem for counting problems due to Creignou and Hermann states that or any finite set S of logical relations, the count- ing problem #SAT(S) is either in FP, or #P-complete. In the present paper we show a dichotomy theorem for polynomial evaluation. That is, we show that for a given set S, either there exists a VNP-complete family of polynomials associated to S, or the associated families of polynomi- als are all in VP. We give a concise characterization of the sets S that give rise to "easy" and "hard" polynomials. We also prove that several problems which were known to be #P-complete under Turing reductions only are in fact #P-complete under many-one reductions.

Foundations of Computational Mathematics, 2003

Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids “unnecessary” branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P$ cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.

Foundations of Computational …, 2005

counting complexity classes #P R and #P C in the Blum-Shub-Smale setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the class FP #P R R . In this paper, we prove that the corresponding result is true over C, namely that the computation of the Euler characteristic of an affine or projective complex variety is complete in the class FP #P C C . We also obtain a corresponding completeness result for the Turing model. * The results of this work were announced in Comptes rendus de l'Académie des sciences Paris,

Theoretical Computer Science, 1978

In the present work we study a proper subclass (the polynomial functions) of the diophant~ne functions and obtain some undecidability results for certain problems of injet:twity, surjecfivity, and bijectivity of such functions. Dio9hantine problems constitute one of the oldest studies of Mathematics. Diopaantine functions, and in particular, polynomial function,,., play an important role in Computer Science. Inde, ed, it follows from the work of Davis, Putnam, Robinson [3] and Matijasevic [6], that a function is diophantine if and only if it is partial recursive. Thus, if we asst,me the availability of an arlgitrary amount of time and memor% the class of diophantine functions is precisely the class of functions calculable by computers. The view that partial recur,Aveness constitutes calculability has another ,mportant eonscq:,ence, in that the, family of undecidable problems cor.stitutes an unattainable upper bound of the capability of the computer. In this sense, any undeeidabl~ result has a kind of "metacomputational" consequence in Computer Science. Decidability, that is, the existence of "effective" procedures or algorithms has been o,'; mu(h concern in the devele, pments of some mathematical theories. ~he invest~.gation through several centuries of a general solution of the algebraic e,:|~tations ef deg~'ee n in one variable, proved fruitless b} a result of E. Galois, is ~hc search of a conexistent algorithm of the simplest type (a formula); in some sense i: was ar~ undecidable question. More recently, Matijasevic [6] has shown theft the prol~lem ef determining of an arbitrarily given polynomial over ':he iatc~cr',. whether o: t~tot it has integral roots is undecidable, i.e., there i,s no general algorithm to 6,ete, rmim', the existence of integral roots. This settled the famous Tenth Prob!cn~ of Hilber'[. A very lucid account of the solulion of this problem is given in [2].

Journal of Computer and System Sciences, 1977

We show that certain problems involving sparse polynomials with integer coefficients are at least as hard as any problem in NP. These problems include determining the degree of the least common multiple of a set of such polynomials, and related problems. The proofs make use of a homomorphism from Boolean expressions over the predicate symbols (P1 ,..-, P-) onto divisors of the polynomial x N-1, where N is the product of the first n primes. Various combinatorial and number theoretic applications are also presented. Two classes of problems have received much attention recently [1-3]. They are the class P of problems solvable in polynomial time by a deterministic Turing machine and the class NP of problems solvable in polynomial time by a nondeterministic Turing machine. It is not known whether P ~ NP. The class NI" contains many interesting combinatorial problems for which no algorithm of less than exponential time is currently known. In this paper we present new results concerning the reducibility of problems in NP to other problems. We say a problem PI is polynomial reducible to a problem P2 if there is an algorithm for solving P1 that spends all but a polynomial amount of its time solving instances of P2. Also, the number of instances of P2 solved must be polynomial in the length of the input. Note that if P2 is in the class P, so is P1. Define NPR to be the class of problems to which any problem in NP is polynomial reducible. Thus, NPR is the set of NP-hard problems. Note that P-NP if any element of NPR is in P. We show that certain problems involving complex polynomials are in the class NPR. It is not known whether these problems are in the class NP. To show that a problem P1 is in NPR, we show that the tautology detection problem Taut is polynomial reducible to P1. It is well known that any element of NP is polynomial reducible to Taut. In this way we show that the following problems are in NPR. (Assume that all polynomials mentioned below are sparse complex polynomials of one variable with real integer coefficients. A sparse polynomial is a polynomial represented in such a way that terms with coefficients of zero are omitted.) P1. Given a finite set of polynomials, to determine (a) Its least common multiple (lcm). (b) The degree of its least common multiple.

1997

Le Centre pour la Communication Scientifique Directe - HAL - Archive ouverte HAL, 2019

In a previous article we prove the Polynomial Hierarchy collapses by making use of a method based essentially on Gaussian Elimination. In this paper we replace completely our use of Gaussian Elimination and formulate an equivalent approach based on the method of Substitution. We give the usual argument that a non-trivial kernel for Exact Satisfiability may be found. Our proof shows the structure formerly known as the Polynomial Hierarchy collapses to the level above P = N P. That is, we show that coNP ⊆ NP \ P.

Mathematical Logic Quarterly, 1998

Motivated by results on interactive proof systems we investigate an 3-V-hierarchy over P using word quantifiers as well as two types of set quantifiers. This hierarchy, which extends the (arithmetic) polynomial-time hierarchy, is called the analytic polynomial-time hierarchy. It is shown that every class of this hierarchy coincides with one of the following Classes: El;, II; (k 2 O) , PSPACE, C y p or IIyp (k 2 1). This improves previous results by Orponen [6] and allows interesting comparisons with the above mentioned results on interactive proof systems.

Journal of Complexity, 2000

We exhibit a new method for showing lower bounds for time-space tradeoffs of polynomial evaluation procedures given by straight-line programs. From the tradeoff results obtained by this method we deduce lower space bounds for polynomial evaluation procedures running in optimal nonscalar time. Time, denoted by L, is measured in terms of nonscalar arithmetic operations and space, denoted by S, is measured by the maximal number of pebbles (registers) used during the given evaluation procedure. The time-space tradeoff function considered in this paper is LS 2 . We show that for``almost all'' univariate polynomials of degree at most d our time-space tradeoff functions satisfy the inequality LS 2 d 8 . From this we conclude that for``almost all'' degree d univariate polynomials, any nonscalar time optimal evaluation procedure requires space at least S c 4 -d, where c>0 is a suitable universal constant. The main part of this paper is devoted to the exhibition of specific families of univariate polynomials which are``hard to compute'' in the sense of time-space tradeoff (this means that our tradeoff function increases linearly in the degree).

The Collected Papers of Stephen Smale, 2000

When complexity theory is studied over an arbitrary unordered eld K , the classical theory is recaptured with K = Z 2 . The fundamental result that the Hilbert Nullstellensatz as a decision problem is NP-complete over K allows us to reformulate and investigate complexity questions within an algebraic framework and to develop transfer principles for complexity theory.