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144
Multivariate Polynomials, Duality, and Structured Matrices
 J. of Complexity
, 1999
"... We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlat ..."
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Cited by 58 (33 self)
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eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving
Polyhedral End Games for Polynomial Continuation
 Numerical Algorithms
, 1998
"... Bernshtein's theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (C ) n , with C = C n f0g. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in ( ..."
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Cited by 22 (8 self)
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Bernshtein's theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (C ) n , with C = C n f0g. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions
Accelerated solution of multivariate polynomial systems of equations
 SIAM Journal on Computing
, 2000
"... Abstract. We propose new Las Vegas randomized algorithms for the solution of a square nondegenerate system of equations, with wellseparated roots. The algorithms use O(δ 3 n D 2 log(D) log(b)) arithmetic operations (in addition to the operations required to compute the normal form of the boundary m ..."
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Cited by 7 (2 self)
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monomials modulo the ideal) to approximate all real roots of the system as well as all roots lying in a fixed ndimensional box or disc. Here D is an upper bound on the number of all complex roots of the system (e.g., Bezout or Bernshtein bound), δ is the number of real roots or the roots lying in the box
Robust SampledData Stabilization of Linear Systems: An Input Delay Approach
 AUTOMATICA
, 2004
"... ..."
The BKK Root Count in C n
 Math. Comp
, 1996
"... Abstract. The root count developed by Bernshtein, Kushnirenko and Khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus (C ∗ ) n. In this paper, we modify this bound slightly so that it counts the number of isolated zeros in C n. Our bound is, apparently, ..."
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Cited by 19 (0 self)
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Abstract. The root count developed by Bernshtein, Kushnirenko and Khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus (C ∗ ) n. In this paper, we modify this bound slightly so that it counts the number of isolated zeros in C n. Our bound is, apparently
The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations
, 2001
"... The mean number of roots of a multihomogeneous system of polynomial equations (with respect to a natural distribution on the space of coefficient vectors) is shown to be at least as large as the square root of the generic number of complex roots, as determined by Bernshtein's [Ber75] theorem. W ..."
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Cited by 17 (2 self)
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The mean number of roots of a multihomogeneous system of polynomial equations (with respect to a natural distribution on the space of coefficient vectors) is shown to be at least as large as the square root of the generic number of complex roots, as determined by Bernshtein's [Ber75] theorem
A Quantum Bit Commitment Scheme Provably Unbreakable by both Parties
, 1993
"... Assume that a party, Alice, has a bit x in mind, to which she would like to be committed toward another party, Bob. That is, Alice wishes, through a procedure commit(x), to provide Bob with a piece of evidence that she has a bit x in mind and that she cannot change it. Meanwhile, Bob should not be ..."
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Cited by 82 (14 self)
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Assume that a party, Alice, has a bit x in mind, to which she would like to be committed toward another party, Bob. That is, Alice wishes, through a procedure commit(x), to provide Bob with a piece of evidence that she has a bit x in mind and that she cannot change it. Meanwhile, Bob should not be able to tell from that evidence what x is. At a later time, Alice can reveal, through a procedure unveil(x), the value of x and prove to Bob that the piece of evidence sent earlier really corresponded to that bit. Classical bit commitment schemes (by which Alice's piece of evidence is classical information such as a bit string) cannot be secure against unlimited computing power and none have been proven secure against algorithmic sophistication. Previous quantum bit commitment schemes (by which Alice's piece of evidence is quantum information such as a stream of polarized photons) were known to be invulnerable to unlimited computing power and algorithmic sophistication, but not to arbitrary...
Intrinsic Near Quadratic Complexity Bounds for Real Multivariate Root Counting
 Proceedings of the Sixth Annual European Symposium on Algorithms, Lecture Notes in Computer Science 1461, SpringerVerlag
, 1998
"... We give a new algorithm, with three versions, for computing the number of real roots of a system of n polynomials equations... ..."
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Cited by 4 (4 self)
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We give a new algorithm, with three versions, for computing the number of real roots of a system of n polynomials equations...
and
, 2009
"... The history of the quadratic stochastic operators can be traced back to the work of S. Bernshtein (1924). For more than 80 years this theory has been developed and many papers were published. In recent years it has again become of interest in connection with numerous applications in many branches of ..."
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The history of the quadratic stochastic operators can be traced back to the work of S. Bernshtein (1924). For more than 80 years this theory has been developed and many papers were published. In recent years it has again become of interest in connection with numerous applications in many branches
Toward a usable theory of Chernoff Bounds for heterogeneous and partially dependent random variables
, 1992
"... Let X be a sum of real valued random variables and have a bounded mean E[X]. The generic ChernoffHoeffding estimate for large deviations of X is: P rfX \GammaE[X ] ag min 0 e \Gamma(a+E[X]) E[e X ], which applies with a 0 to random variables with very small tails. At issue is how to use this ..."
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Cited by 7 (1 self)
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Let X be a sum of real valued random variables and have a bounded mean E[X]. The generic ChernoffHoeffding estimate for large deviations of X is: P rfX \GammaE[X ] ag min 0 e \Gamma(a+E[X]) E[e X ], which applies with a 0 to random variables with very small tails. At issue is how to use
Results 1  10
of
144