pinecone/core-2020-05-10-deduplication huggingface dataset:
Deformation Techniques for Efficient Polynomial Equation Solving

AbstractSuppose we are given a parametric polynomial equation system encoded by an arithmetic circuit, which represents a generically flat and unramified family of zero-dimensional algebraic varieties. Let us also assume that there is given the complete description of the solution of a particular unramified parameter instance of our system. We show that it is possible to “move” the given particular solution along the parameter space in order to reconstruct—by means of an arithmetic circuit—the coordinates of the solutions of the system for an arbitrary parameter instance. The underlying algorithm is highly efficient, i.e., polynomial in the syntactic description of the input and the following geometric invariants: the number of solutions of a typical parameter instance and the degree of the polynomials occurring in the output. In fact, we prove a slightly more general result, which implies the previous statement by means of a well-known primitive element algorithm. We produce an efficient algorithmic description of the hypersurface obtained projecting polynomially the given generically flat family of varieties into a suitable affine space
Multivariate Polynomials, Duality, and Structured Matrices

AbstractWe 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 correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues, and relations between them are studied. Finally, we show some applications of this study to rootfinding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians, and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to the matrix 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 multivariate polynomial systems of equations
Fast Computation of Contour Integrals of Rational Functions

AbstractThis paper presents fast algorithms for computing numerical approximations for contour integrals of rational functions. Given the coefficients of two polynomials q and p∈C[z], a curve Γ in the complex plane, and an error bound ε, the integral ∫Γq(z)/p(z)dz is computed up to an error of ε. In the special case that the zeros of p lie in a small disc not intersected by Γ, the integral is computed by summing up the integrals of an initial segment of a suitable Laurent series of q/p. The general case is reduced to this special one by partial fraction decomposition as described by P. Kirrinnis (1998, Partial fraction decomposition in C(z) and simultaneous Newton iteration for factorization in C[z], J. Complexity14, 378–444. The algorithms are analyzed from the point of view of (serial) bit complexity. The running time of the algorithms is estimated in terms of the error bound prescribed for the result, the degree of the polynomials involved, and the condition of the problem, measured by a lower bound for the distance between the zeros of p and the points of Γ. This condition parameter need not be known in advance
Approximating Complex Polynomial Zeros: Modified Weyl's Quadtree Construction and Improved Newton's Iteration

AbstractWe propose a new algorithm for the classical and still practically important problem of approximating zeros zj of an nth degree polynomial p(x) within error bound 2−bmaxj|zj|. The algorithm uses O((n2logn)log(bn)) arithmetic operations and comparisons for approximating all the n zeros and O((knlogn)log(bn)) for approximating the k zeros lying in a fixed domain (disc or square) and isolated from the other zeros. Unlike the previous fast algorithms of this kind, the new algorithm has its simple elementary description, is convenient for practical implementation, and allows the users to adapt the computational precision to the current level of approximation achieved in the process of computing and ultimately to the requirements to the output precision for each zero of p(x). The algorithm relies on our novel versions of Weyl's quadtree construction and Newton's iteration
Lifting/Descending Processes for Polynomial Zeros

AbstractThe recently proposed Chebyshev-like lifting map for the zeros of a univariate polynomial was motivated by its applications to splitting a univariate polynomial p(z) numerically into factors, which is a major step of some most efficient algorithms for approximating polynomial zeros. We complement the Chebyshev-like lifting process by a descending process, decrease the estimated computational cost of performing the algorithm, demonstrate its correlation to Graeffe's lifting/descending process, and generalize lifting from Graeffe's and Chebyshev-like maps to any fixed rational map of the zeros of the input polynomial
On Symbolic-Numeric Solving of Sine-Polynomial Equations

AbstractThis paper deals with the localization of all the real roots of sine-polynomials. D. Richardson (1991, in “effective Methods in Algebraic Geometry,” pp. 427–440, Birkhäuser, Basel) has already studied this type of analytic function. He showed how to find the number of real roots in a bounded interval. Here we propose an algorithm which determines whether a sine-polynomial has a finite number of real roots or not. Moreover in the finite case we construct an explicit interval containing all of them. We also construct a generalized exclusion method to find all the real roots in a bounded interval and we adapt it to the infinite case
Complexity of Linear Problems with a Fixed Output Basis

AbstractWe use an information-based complexity approach to study the complexity of approximation of linear operators. We assume that the values of a linear operator may be elements of an infinite dimensional space G. It seems reasonable to look for an approximation as a linear combination of some elements gi from the space G and compute only the coefficients ci of this linear combination. We study the case when the elements gi are fixed and compare it with the case where the gi can be chosen arbitrarily. We show examples of linear problems where a significant output data compression is possible by the use of a nonlinear algorithm, and this nonlinear algorithm is much better than all linear algorithms. We also provide an example of a linear problem for which one piece of information is enough whereas an optimal (minimal cost) algorithm must use information of much higher cardinality
What Is the Complexity of Stieltjes Integration?

AbstractWe study the complexity of approximating the Stieltjes integral ∫10f(x)dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(ε) denote the ε-complexity (the minimal cost of computing an ε-approximation). We show that r(n)≍n−min{r, s+1} and that comp(ε)≍ε−1/min{r, s+1}. We also present an algorithm that computes an ε-approximation at nearly minimal cost
Decision Problems in Quadratic Function Fields of High Genus

AbstractThis paper provides verification procedures for a number of decision problems in quadratic function fields of odd characteristic, thereby establishing membership of these problems in both NP and co-NP. The problems include determining the ideal and divisor class numbers of the field, the regulator of the field (in the real case), a generating system of the ideal class group, a basis of the ideal class group, the pricipality of an ideal, the equivalence of two ideals, the discrete logarithm of an ideal class with respect to another ideal class, and the order of a class in the ideal class group. While several of these problems belong to the aforementioned complexity classes unconditionally, others require a certain assumption to ensure that the verification procedures can be done in polynomial time; so far, this assumption has only been verified for fields of high genus
On the Communication Complexity of Lipschitzian Optimization for the Coordinated Model of Computation

AbstractWe consider the problem of approximating the maximum of the sum of m Lipschitz continuous functions. The values of each function are assumed to reside at a different memory element. A single processing element is designated to approximate the value of the maximum of the sum of these functions by adopting a certain protocol. Under certain assumptions on the class of permissible protocols, we obtain the minimum number of real-valued messages that has to be transferred between the processing element and the memory elements in order to find the desired approximation of this maximum. In particular, we exploit the optimality of the nonadaptive protocols for the Lipschitzian optimization problem, studied in the context of information-based complexity, to prove our main result
Delayed Curse of Dimension for Gaussian Integration

AbstractWe study Gaussian integration of Lipschitz functions in the ℓp-norm, p∈[1, ∞], defined over the d dimensional space Rd. It is assumed that the Gaussian weight has a diagonal covariance matrix with variances σi. We consider the problem of how the worst case complexity of computing an ε-approximation, comp(ε), depends on the variances σi. If ε goes to zero then comp(ε) is of order ε−d and we have the curse of dimension. For a fixed ε we provide lower and upper bounds on comp(ε) of the form ε−t where the exponent t is small for small variances σi. We call this moderation of the exponential growth of the complexity, the delayed curse of dimension. We prove similar upper and lower bounds for the integration of Lipschitz functions on sequence spaces ℓp, p∈[1, ∞), which is essentially the problem of path integration with a Gaussian measure for which ∑∞i=1σp/2i<∞. Finally, we show how the same proof technique yields an upper bound when the integrand belongs to the Sobolev class Wr, ∞(Rd), d<∞, of functions with bounded (generalized) derivatives of order r, and when Hermite information is used. This upper bound shows that the moderating effect of the variances becomes much more essential for large r
Some Speed-Ups and Speed Limits for Real Algebraic Geometry

AbstractWe give new positive and negative results, some conditional, on speeding up computational algebraic geometry over the reals: 1.A new and sharper upper bound on the number of connected components of a semi-algebraic set. Our bound is novel in that it is stated in terms of the volumes of certain polytopes and, for a large class of inputs, beats the best previous bounds by a factor exponential in the number of variables. 2.A new algorithm for approximating the real roots of certain sparse polynomial systems. Two features of our algorithm are (a) arithmetic complexity polylogarithmic in the degree of the underlying complex variety (as opposed to the super-linear dependence in earlier algorithms) and (b) a simple and efficient generalization to certain univariate exponential sums. 3.Detecting whether a real algebraic surface (given as the common zero set of some input straight-line programs) is not smooth can be done in polynomial time within the classical Turing model (resp. BSS model over C) only if P=NP (resp. NP⊆BPP). The last result follows easily from an unpublished observation of S. Smale
Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

AbstractMany applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem
Finding a Cluster of Zeros of Univariate Polynomials

AbstractA method to compute an accurate approximation for a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newton's method, and Rouché's theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of the Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. In this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster. Finally, an inclusion test based on a punctual version of the Rouché theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed
Iteration, Inequalities, and Differentiability in Analog Computers

AbstractShannon's general purpose analog computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x)∈G there is a function F(x, t)∈G such that F(x, t)=ft(x) for nonnegative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions xkθ(x) that sense inequalities in a differentiable way, the resulting class, which we call G+θk, is closed under iteration. Furthermore, G+θk includes all primitive recursive functions and has the additional closure property that if T(x) is in G+θk, then any function ofx computable by a Turing machine in T(x) time is also
Multi-sequences with Almost Perfect Linear Complexity Profile and Function Fields over Finite Fields

AbstractSequences with almost perfect linear complexity profile defined by Niederreiter (1997, Lecture Notes in Computer Science, Vol. 304, pp. 37–51, Springer-Verlag, Berlin/New York) are quite important for stream ciphers. In this paper, we investigate multi-sequences with almost perfect linear complexity profile and obtain a construction of such multi-sequences by using function fields over finite fields. Some interesting examples from this construction are presented to illustrate our construction
Weakly Computable Real Numbers

AbstractA real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi-computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there is a computable sequence (xs)s∈N of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers
Computation of Affine Covariants of Quadratic Bivariate Differential Systems

AbstractThis paper deals with affine covariants of autonomous differential systems. The main result is the construction of a minimal system of generators of the algebra of affine covariants of quadratic bivariate differential systems which is helpful in qualitative and numerical study. To this end, we establish a theorem (true for general systems of dimension n and degree m) which provides a procedure of construction of systems of generators for affine covariants from those of center-affine invariants. After applying this theorem to the case n=m=2 we give the expansions of the obtained affine covariants in terms of center-affine covariants. All algorithms constitute the package SIB
Quantum Complexity of Integration

AbstractIt is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes Fk, αd on [0, 1]d and define γ by γ=(k+α)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are comp(Fk, αd, ε)≍ε−1/γ and comprandom(Fk, αd, ε)≍ε−2/(1+2γ). For a quantum computer we prove compquantquery(Fk, αd, ε)≍ε−1/(1+γ) and compquant(Fk, αd, ε)⩽Cε−1/(1+γ)(logε−1)1/(1+γ). For restricted Monte Carlo (only coin tossing instead of general random numbers) we prove compcoin(Fk, αd, ε)⩽Cε−2/(1+2γ)(logε−1)1/(1+2γ). To summarize the results one can say that •there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if γ is small; •there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical methods, if γ is small
On the Isomorphism of Fractional Factorial Designs

AbstractTwo fractional factorial designs are called isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and switching the levels of factors. To identify the isomorphism of two s-factor n-run designs is known to be an NP hard problem, when n and s increase. There is no tractable algorithm for the identification of isomorphic designs. In this paper, we propose a new algorithm based on the centered L2-discrepancy, a measure of uniformity, for detecting the isomorphism of fractional factorial designs. It is shown that the new algorithm is highly reliable and can significantly reduce the complexity of the computation. Theoretical justification for such an algorithm is also provided. The efficiency of the new algorithm is demonstrated by using several examples that have previously been discussed by many others
The Optimal Discretization of Stochastic Differential Equations

AbstractWe study pathwise approximation of scalar stochastic differential equations. The mean squared L2-error and the expected number n of evaluations of the driving Brownian motion are used for the comparison of arbitrary methods. We introduce an adaptive discretization that reflects the local properties of every single trajectory. The corresponding error tends to zero like c·n−1/2, where c is the average of the diffusion coefficient in space and time. Our method is justified by the matching lower bound for arbitrary methods that are based on n evaluations on the average. Hence the adaptive discretization is asymptotically optimal. The new method is very easy to implement, and about 7 additional arithmetical operations are needed per evaluation of the Brownian motion. Hereby we can determine the complexity of pathwise approximation of stochastic differential equations. We illustrate the power of our method already for moderate accuracies by means of a simulation experiment
A Gröbner Free Alternative for Polynomial System Solving

AbstractGiven 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 of the system. We present our implementation in the Magma system which is called Kronecker in homage to his method for solving systems of polynomial equations. We show that the theoretical complexity of our algorithm is well reflected in practice and we exhibit some cases for which our program is more efficient than the other available software
Kronecker's and Newton's Approaches to Solving: A First Comparison

AbstractThese pages are a first attempt to compare the efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker's solution (from the symbolic approach) with approximate zero theory (introduced by S. Smale as a foundation of numerical analysis). For this purpose we show upper and lower bounds of the bit length of approximate zeros. We also introduce efficient procedures that transform local Kronecker solutions into approximate zeros and conversely. As an application of our study we exhibit an efficient procedure to compute splitting field and Lagrange resolvent of univariate polynomial equations. We remark that this procedure is obtained by a convenient combination of both approaches (numeric and symbolic) to multivariate polynomial solving