Results 1  10
of
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 ..."
Abstract

Cited by 29 (12 self)
 Add to MetaCart
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
"... . 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 SchonhageStrass ..."
Abstract

Cited by 27 (9 self)
 Add to MetaCart
. 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 SchonhageStrassen trick, Schonhage's trick, Nussbaumer's trick, the cyclic SchonhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms. 1.
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 ..."
Abstract

Cited by 27 (7 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.
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 ..."
Abstract

Cited by 17 (8 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
. 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...
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, ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
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.
ON THE RANKS AND BORDER RANKS OF SYMMETRIC TENSORS
"... Abstract. Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Abstract. Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent. 1.
The border rank of the multiplication of 2 × 2 matrices is seven
 J. Amer. Math. Soc
"... One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and