Results 1  10
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24
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 82 (16 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits
 SIAM J. COMPUT
, 2007
"... In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given ..."
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Cited by 26 (7 self)
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In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if E: F n ↦ → F m is a linear LDC with two queries, then m = exp(Ω(n)). Previously this was known only for fields of size ≪ 2 n [O. Goldreich et al., Comput. Complexity, 15 (2006), pp. 263–296]. (2) We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fanin that computes the zero polynomial, one can construct an LDC. More formally, assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear functions appearing in C. Then we can construct a linear LDC with two queries that encodes messages of length r/polylog(d) by codewords of length O(d). (3) We prove a structural theorem for ΣΠΣ circuits, with a bounded top fanin, that
Linear recurrences with polynomial coefficients and computation of the CartierManin operator on hyperelliptic curves
 In International Conference on Finite Fields and Applications (Toulouse
, 2004
"... Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of h ..."
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Cited by 21 (8 self)
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Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.
Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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Cited by 21 (13 self)
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
On The Multiplicative Complexity of Boolean Functions over the Basis ...
, 1998
"... . The multiplicative complexity c(f) of a Boolean function f is the minimum number of AND gates in a circuit representing f which employs only AND, XOR and NOT gates. A constructive upper bound, c(f) = 2 n 2 +1 \Gamma n=2 \Gamma 2, for any Boolean function f on n variables (n even) is given. A c ..."
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Cited by 17 (6 self)
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. The multiplicative complexity c(f) of a Boolean function f is the minimum number of AND gates in a circuit representing f which employs only AND, XOR and NOT gates. A constructive upper bound, c(f) = 2 n 2 +1 \Gamma n=2 \Gamma 2, for any Boolean function f on n variables (n even) is given. A counting argument gives a lower bound of c(f) = 2 n 2 \Gamma O(n). Thus we have shown a separation, by an exponential factor, between worstcase Boolean complexity (which is known to be \Theta(2 n n \Gamma1 )) and worstcase multiplicative complexity. A construction of circuits for symmetric Boolean functions on n variables, requiring less than n + 3 p n AND gates, is described. 1 Introduction. A fair amount of research in Boolean circuit complexity is devoted to the following problem: Given a Boolean function and a supply of gates that perform certain basic operations, construct a circuit which corresponds (in some way) to the function and is optimal (in some sense). A wellstudied...
Lower Bounds and Separations for Constant Depth Multilinear Circuits
"... We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a superpolynomial separation between ..."
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Cited by 15 (6 self)
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We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a superpolynomial separation between the size of productdepth 1 d and productdepth d + 1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that • There exists a multilinear circuit of productdepth d + 1 and of polynomial size computing f. • Every multilinear circuit of productdepth d computing f has superpolynomial size. 1
A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
 Proceedings of the 48th FOCS: 438–448
, 2007
"... We construct an explicit polynomial f(x1,...,xn), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Ω(n 4/3 / log 2 n). The lower bound holds over any field. ..."
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Cited by 13 (9 self)
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We construct an explicit polynomial f(x1,...,xn), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Ω(n 4/3 / log 2 n). The lower bound holds over any field.
Variations by complexity theorists on three themes of
 Computational Complexity
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 12 (4 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways. 1
The Intrinsic Complexity of Parametric Elimination Methods
, 1998
"... This paper is devoted to the complexity analysis of a particular property, called geometric robustness owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which owns this property must necess ..."
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Cited by 10 (5 self)
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This paper is devoted to the complexity analysis of a particular property, called geometric robustness owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which owns this property must necessarily have an exponential sequential time complexity even if highly performant data structures (as e.g. the straightline program encoding of polynomials) are used. The paper finishes with the motivated introduction of a new nonuniform complexity measure for zerodimensional polynomial equation systems, called elimination complexity.
Modular Algorithms for Polynomial Basis Conversion and Greatest Factorial Factorization
, 2000
"... We give new algorithms for converting between representations of polynomials with respect to certain kinds of bases, comprising the usual monomial basis and the falling factorial basis, for fast multiplication and Taylor shift in the falling factorial basis, and for computing the greatest factori ..."
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Cited by 9 (0 self)
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We give new algorithms for converting between representations of polynomials with respect to certain kinds of bases, comprising the usual monomial basis and the falling factorial basis, for fast multiplication and Taylor shift in the falling factorial basis, and for computing the greatest factorial factorization. We analyze both the classical and the new algorithms in terms of arithmetic coefficient operations. For the special case of polynomials with integer coefficients, we present modular variants of these methods and give cost estimates in terms of word operations.