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A Deterministic Algorithm to Compute Approximate Roots of Polynomial Systems in Polynomial Average Time

Pierre Lairez
2016 Foundations of Computational Mathematics  
We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum-Shub-Smale  ...  It rests upon a derandomization of an algorithm of Beltr\'an and Pardo and gives a deterministic affirmative answer to Smale's 17th problem.  ...  Acknowledgment I am very grateful to Peter Bürgisser for his help and constant support, and to Carlos Beltrán for having carefully commented this work.  ... 
doi:10.1007/s10208-016-9319-7 fatcat:6pwhizjagzfbhf6llvdiqvxdly

Foreword

Teresa Krick, James Renegar
2015 Foundations of Computational Mathematics  
Mike and Steve conjectured that on average, a root of a polynomial system can be approximated accurately in polynomial time, a conjecture finally settled affirmatively by Carlos Beltrán and Luis Pardo  ...  In the early 1980s, Mike began collaborating with Steve in investigating the computational complexity of algorithms for approximating roots of polynomials, initially for univariate polynomials and later  ... 
doi:10.1007/s10208-014-9234-8 fatcat:bjau57ao6neb5muehdh6lm6ppq

Page 489 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
As evidence toward this conjecture, we show that if any polynomial-time algorithm can approximate the longest path to a ratio of 29°“), for any e > 0, then NP has a quasi-polynomial deterministic time  ...  We then show that no polynomial- time algorithm can find a constant factor approximation to the longest-path problem unless P = NP.  ... 

Deterministic wavelet thresholding for maximum-error metrics

Minos Garofalakis, Amit Kumar
2004 Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems - PODS '04  
We introduce an optimal low polynomial-time algorithm for one-dimensional wavelet thresholding -our algorithm is based on a new Dynamic-Programming (DP) formulation, and can be employed to minimize the  ...  Unfortunately, directly extending our one-dimensional DP algorithm to multi-dimensional wavelets results in a super-Permission to make digital or hard copies of all or part of this work for personal or  ...  We use this idea to devise a polynomial-time (1 + )-approximation scheme. Given a threshold parameter τ > 0, we define a truncated DP algorithm as follows.  ... 
doi:10.1145/1055558.1055582 dblp:conf/pods/GarofalakisK04 fatcat:3run2fuxp5bxxawt4aoqrhqzbe

An Explicit VC-Theorem for Low-Degree Polynomials [chapter]

Eshan Chattopadhyay, Adam Klivans, Pravesh Kothari
2012 Lecture Notes in Computer Science  
We also give the first hardness result for this problem and show that the existence of a poly(n k , |X|, 1/ǫ)-time algorithm for deterministically constructing ǫ-approximations for circuits of size n k  ...  We show that for any X ⊆ R n and any Boolean function class C that is uniformly approximated by degree k, low-weight polynomials, an ǫ-approximation S can be be constructed deterministically in time poly  ...  Hardness of Constructing ǫ-Approximations In this section we show that an efficient deterministic algorithm to compute an ǫ-approximation (or a hard function) for the class of polynomial size circuits  ... 
doi:10.1007/978-3-642-32512-0_42 fatcat:qef3ghouf5dihm6ttda42b7efe

A Bluff-and-Fix Algorithm for Polynomial Chaos Methods [chapter]

Laura Lyman, Gianluca Iaccarino
2020 Lecture Notes in Computer Science  
The strategy is to express the resulting stochastic solution using M + 1 terms of a polynomial chaos expansion and then derive and solve a deterministic, coupled system of PDEs with standard numerical  ...  Stochastic Galerkin methods can be used to approximate the solution to a differential equation in the presence of uncertainties represented as stochastic inputs or parameters.  ...  This representation has its roots in the work of Wiener [7] , who expressed a Gaussian process as an infinite series of Hermite polynomials.  ... 
doi:10.1007/978-3-030-50436-6_55 fatcat:2kpxtgdm7fehpdaunv24lwcebe

The Module Structure of ARMAX Systems

Ivan Markovsky, Jan C. Willems, Bart De Moor
2006 Proceedings of the 45th IEEE Conference on Decision and Control  
We propose an ARMAX identification algorithm, which has three steps: first compute the deterministic part of the system via the orthogonalizers, then the AR part, which also has a module structure, and  ...  Computing a module basis for the orthogonalizers is a deterministic identification problem.  ...  polynomial time algorithms of [7] or [8] .  ... 
doi:10.1109/cdc.2006.377656 dblp:conf/cdc/MarkovskyWM06 fatcat:ojmeavucp5g3jcguwqbukgigpm

A Faster Solution to Smale's 17th Problem I: Real Binomial Systems [article]

Grigoris Paouris, Kaitlyn Phillipson, J. Maurice Rojas
2019 arXiv   pre-print
We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just O(n^2((n)+_i d_i)) arithmetic operations on average.  ...  We also discuss briefly the obstructions to maintaining average-case time polynomial in n_i d_i when F has more terms.  ...  ACKNOWLEDGEMENTS We humbly thank REU students Caleb Bugg and Paula Burkhardt for important discussions on preliminary versions of this work.  ... 
arXiv:1901.09739v1 fatcat:k3k4hubsq5c23f4syikndvkqdq

Four Integer Factorization Algorithms [article]

N. A. Carella
2010 arXiv   pre-print
The running time complexity of these algorithms ranges from deterministic exponential time complexity O(N^(1/2)) to heuristic and unconditional logarithmic time complexity O((log N)^c), c > 0 constant.  ...  The theoretical aspects of four integer factorization algorithms are discussed in details in this note.  ...  On average, the norm of a small vector in a lattice is approximated by the geometric mean, that is, n n i L V / 1 2 / 1 2 ) det( γ ≤ .  ... 
arXiv:1003.3261v3 fatcat:hxqhnl7yq5fivjurgr2466hluu

Efficient uncertainty quantification with the polynomial chaos method for stiff systems

Haiyan Cheng, Adrian Sandu
2009 Mathematics and Computers in Simulation  
In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost.  ...  When stiff systems are considered, implicit numerical integration requires the solution of a nonlinear system of equations at every time step.  ...  But at the same time, the computation time is increased due to the increase of the number of the deterministic runs.  ... 
doi:10.1016/j.matcom.2009.05.002 fatcat:4ih3cwechzftbkk64r4w4dno2a

Efficiently Computing the Density of Regular Languages [chapter]

Manuel Bodirsky, Tobias Gärtner, Timo von Oertzen, Jan Schwinghammer
2004 Lecture Notes in Computer Science  
We present an algorithm that computes the number of accumulation points of (fm) in polynomial time, if the regular language L is given by a finite deterministic automaton, and can then also efficiently  ...  A regular language L is called dense if the fraction fm of words of length m over some fixed signature that are contained in L tends to one if m tends to infinity.  ...  To approximate the values at the roots of unity up to n bits we need O(n 2 log n) time.  ... 
doi:10.1007/978-3-540-24698-5_30 fatcat:bukd7lwxa5behbvzvaiyq5saqm

Intrinsic Near Quadratic Complexity Bounds for Real Multivariate Root Counting [chapter]

J. Maurice Rojas
1998 Lecture Notes in Computer Science  
We give a new algorithm, with three versions, for computing the number of real roots of a system of n polynomial equations in n unknowns.  ...  The rst version is of Monte Carlo type and, neglecting logarithmic factors, runs in time quadratic in the average number of complex roots of a closely related system.  ...  A Simple Example of Our Algorithm Suppose one would like to count the number of real roots of the following bivariate polynomial system: f 1 (x; y) = 1 ? 2x 8 y 2 ?  ... 
doi:10.1007/3-540-68530-8_11 fatcat:fdw7eef5h5cbjp2uxljgtqlzy4

On the Complexity of Diophantine Geometry in Low Dimensions [article]

J. Maurice Rojas
1998 arXiv   pre-print
Better still, we show that the truth of the Generalized Riemann Hypothesis implies that detecting roots in Q^n for the polynomial systems in (I) can be done via a two-round Arthur-Merlin protocol, i.e.  ...  A practical point of interest is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems.  ...  So via [Coh93] , a putative root of F mod p can indeed be verified in polynomial time. So we indeed need only one call to an NP oracle. Our algorithm is thus indeed an AM algorithm.  ... 
arXiv:math/9811088v1 fatcat:2p7qxmzr6ra5hjimaajaocqspu

On the complete instability of interval polynomials

F. Dabbene, B.T. Polyak, R. Tempo
2007 Systems & control letters (Print)  
That is, the objective is to check if all polynomials in the family are unstable. If not, a subsequent goal is to find a stable polynomial.  ...  The second contribution of this paper is to provide a probability-one estimate of the volume of stable polynomials. These results are based on a combination of deterministic and randomized methods.  ...  A different approach followed in this literature is to develop efficient polynomial-time randomized algorithms for computing the volume of convex bodies, see [5] .  ... 
doi:10.1016/j.sysconle.2006.11.002 fatcat:wouesjp3uncmxizhkr2n3zn62q

Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs [article]

Marc Lelarge, Hang Zhou
2013 arXiv   pre-print
We show that our algorithm approximates the probability for a vertex to be covered by a matching, sampled according to this Gibbs distribution, in a near-optimal sublinear time.  ...  We extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability, where again the query complexity is polynomial in 1/ϵ and the lower  ...  The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-11-JS02-005-01 (GAP project).  ... 
arXiv:1208.3629v5 fatcat:5ewl7embvbcn5odnhhbbyaqypq
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