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8,082
Polynomial degree vs. quantum query complexity
 Proceedings of FOCS’03
"... The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with pol ..."
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Cited by 81 (14 self)
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The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function
On the Polynomial Degree of MintermCyclic Functions
"... When evaluating Boolean functions, each bit of input that must be checked is costly, so we want to know how many bits must be checked for a given function. To aid in this analysis, a variety of complexity measures exist, providing easier methods of analysis to find lower bounds on the function’s cos ..."
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cost. An important complexity measure is "polynomial degree", defined as the degree of the (unique) multilinear polynomial that represents the Boolean function. We look at the polynomial degrees of mintermcyclic functions, which are a type of patternmatching problem. We prove, for monotone
The polynomial degree of recursive fourier sampling
 In Proceedings of the Fifth Annual Conference on Theory of Quantum Computation, Communication and Cryptography (TQC
, 2010
"... Abstract. We present matching upper and lower bounds for the “weak” polynomial degree of the recursive Fourier sampling problem from quantum complexity theory. The degree bound is h+ 1, where h is the order of recursion in the problem’s definition, and this bound is exponentially lower than the bou ..."
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Cited by 1 (1 self)
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Abstract. We present matching upper and lower bounds for the “weak” polynomial degree of the recursive Fourier sampling problem from quantum complexity theory. The degree bound is h+ 1, where h is the order of recursion in the problem’s definition, and this bound is exponentially lower than
Factoring polynomials with rational coefficients
 MATH. ANN
, 1982
"... In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomia ..."
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Cited by 961 (11 self)
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to be factored, n = deg(f) is the degree of f, and for a polynomial ~ a ~ i with real coefficients a i. i An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a padic irreducible factor h of f, to a certain precision. This is done with Berlekamp's algorithm
Approximate polynomial degree of Boolean functions and its applications
"... The approximate polynomial degree of a Boolean function f: {0, 1} n → {0, 1} is the smallest k such that there is a degreek polynomial approximating f on 0/1 inputs. We survey recent developments on this subject. ..."
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Cited by 1 (0 self)
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The approximate polynomial degree of a Boolean function f: {0, 1} n → {0, 1} is the smallest k such that there is a degreek polynomial approximating f on 0/1 inputs. We survey recent developments on this subject.
The complexity of theoremproving procedures
 IN STOC
, 1971
"... It is shown that any recognition problem solved by a polynomial timebounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology. Here “reduced ” means, roughly speaking, that the first problem can be solved deterministi ..."
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Cited by 1050 (5 self)
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deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first
Robust Preconditioners for DG–Discretizations with Arbitrary Polynomial Degrees
, 2012
"... Discontinuous Galerkin (DG) methods offer an enormous flexibility regarding local grid refinement and variation of polynomial degrees for a variety of different problem classes. With a focus on diffusion problems, we consider DG discretizations for elliptic boundary value problems, in particular the ..."
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Cited by 2 (2 self)
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Discontinuous Galerkin (DG) methods offer an enormous flexibility regarding local grid refinement and variation of polynomial degrees for a variety of different problem classes. With a focus on diffusion problems, we consider DG discretizations for elliptic boundary value problems, in particular
The polynomial degree of the Grassmannian G1;n;2
"... For a subset of PG(N; 2) a known result states that has polynomial degree r; r N; if and only if intersects every rat of PG(N; 2) in an odd number of points. Certain re
nements of this result are considered, and are then applied in the case when is the Grassmannian G1;n;2 PG(N; 2); N = n+1 ..."
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For a subset of PG(N; 2) a known result states that has polynomial degree r; r N; if and only if intersects every rat of PG(N; 2) in an odd number of points. Certain re
nements of this result are considered, and are then applied in the case when is the Grassmannian G1;n;2 PG(N; 2); N = n+1
Freeform deformation of solid geometric models
 IN PROC. SIGGRAPH 86
, 1986
"... A technique is presented for deforming solid geometric models in a freeform manner. The technique can be used with any solid modeling system, such as CSG or Brep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, f ..."
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Cited by 701 (1 self)
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, for example. The deformation can be applied either globally or locally. Local deformations can be imposed with any desired degree of derivative continuity. It is also possible to deform a solid model in such a way that its volume is preserved. The scheme is based on trivariate Bernstein polynomials
An Efficient Solution to the FivePoint Relative Pose Problem
, 2004
"... An efficient algorithmic solution to the classical fivepoint relative pose problem is presented. The problem is to find the possible solutions for relative camera pose between two calibrated views given five corresponding points. The algorithm consists of computing the coefficients of a tenth degre ..."
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Cited by 484 (13 self)
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degree polynomial in closed form and subsequently finding its roots. It is the first algorithm well suited for numerical implementation that also corresponds to the inherent complexity of the problem. We investigate the numerical precision of the algorithm. We also study its performance under noise
Results 1  10
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8,082