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32
Polar varieties and efficient real elimination
 MATH. Z
, 2001
"... Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations th ..."
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Cited by 29 (13 self)
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Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.
Multidigit Multiplication For Mathematicians
, 2001
"... This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStrassen ..."
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Cited by 27 (9 self)
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This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStrassen trick, Schönhage's trick, Nussbaumer's trick, the cyclic SchönhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms.
Generalized polar varieties: Geometry and algorithms
, 2004
"... Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motiva ..."
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Cited by 26 (8 self)
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Let V be a closed algebraic subvariety of the n–dimensional projective space over the complex or real numbers and suppose that V is non–empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in [2]. As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar varietiy, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in
Fast Multiplication And Its Applications
"... This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time. ..."
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Cited by 20 (4 self)
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This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.
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...
Separation of Multilinear Circuit and Formula Size
 Theory of Computing
, 2006
"... Abstract: An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit polynomial f (x1,...,xn) with coefficients in {0,1} such that over any field: 1. f can be computed by a polynomialsize multilinear circuit of depth O(log 2 n ..."
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Cited by 17 (8 self)
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Abstract: An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit polynomial f (x1,...,xn) with coefficients in {0,1} such that over any field: 1. f can be computed by a polynomialsize multilinear circuit of depth O(log 2 n). 2. Any multilinear formula for f is of size n Ω(logn). This gives a superpolynomial gap between multilinear circuit and formula size, and separates multilinear NC1 circuits from multilinear NC2 circuits. ACM Classification: F.2.2, F.1.3, F.1.2, G.2.0
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 14 (2 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Elusive functions and lower bounds for arithmetic circuits
 Electronic Colloquium in Computational Complexity
, 2007
"... A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affin ..."
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Cited by 13 (2 self)
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A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f: C → Cm, such that, the image of f is not contained in the image of any polynomialmapping Γ: Cm−1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness2), of degree up to 2mo(1) , implies superpolynomial lower bounds for computing the permanent over C. More generally, we say that a polynomialmapping f: Fn → Fm is (s, r)elusive, if for every polynomialmapping Γ: Fs → Fm of degree r, Image(f) � ⊂ Image(Γ).
Asymptotically fast group operations on Jacobians of general curves
 Mathematics of Computation
, 2007
"... Abstract. Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of ..."
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Cited by 11 (1 self)
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Abstract. Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of dimension at most O(g log g), and so take O(g 3+ɛ) field operations in k, using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to O(g 2.376). This represents a significant improvement over the previous record of O(g 4) field operations (also after a precomputation) for general curves of genus g. 1.
Geometric complexity theory and tensor rank (Extended Abstract)
, 2011
"... Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tens ..."
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Cited by 10 (4 self)
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Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G = GL(W1) × GL(W2) × GL(W3) acting on the tensor product W = W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs = SL(W1) × SL(W2) × SL(W3). A key idea from [26] is that the irreducible Gsrepresentations occurring in the coordinate ring of the Gorbit closure of as table tensor w∈W are exactly those having a nonzero invariant with respect to the stabilizer group of w. However, we prove that by considering Gsrepresentations, only trivial lower bounds on border rank can be shown. It is thus necessary to study Grepresentations, which leads to geometric extension problems that are beyond the scope of the subgroup restriction problems emphasized in [25, 26] and its follow up papers. We prove a very modest lower bound on the border rank of matrix multiplication tensors using Grepresentations. This shows at least that the barrier for Gsrepresentations can be overcome. To advance, we suggest the coarser approach to replace the semigroup of representations of a tensor by its moment polytope. We prove first results towards determining the moment polytopes of matrix multiplication and unit tensors.