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Complexities Approach to Two Problems In Number Theory [article]

Yang Bai, Xiuli Wang
2016 arXiv   pre-print
By Kolmogorov Complexity,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 Exhibit at least one number which does  ...  problems in the non-logical discipline.Futhermore,we show that resource-bounded Kolmogorov Complexity and computational complexity can at least provide tips or principles to mathematical problems in the  ...  complexity is of the algorithm which is the most efficient one.  ... 
arXiv:1610.04026v3 fatcat:5udl5eurnzh63c6ysh6b7kta6y

Algorithms in Real Algebraic Geometry: A Survey [article]

Saugata Basu
2014 arXiv   pre-print
We also describe some recent results linking the computational hardness of decision problems in the first order theory of the reals, with that of computing certain topological invariants of semi-algebraic  ...  We emphasize throughout the complexity aspects of these algorithms and also discuss the computational hardness of the underlying problems.  ...  of the reals with a fixed number of quantifiers to the problem of computing Betti numbers of semi-algebraic sets.  ... 
arXiv:1409.1534v1 fatcat:nyprfglktvdtnmhu3zwqrb547y

Computational Real Algebraic Geometry [chapter]

Bhubaneswar Mishra
2004 Handbook of Discrete and Computational Geometry, Second Edition  
1) 2 : If, on the other hand, the underlying domain is assumed to be the eld of complex numbers (an algebraically closed eld), then other tools from computational algebra is used (e. Tar51] .  ...  By convention, the degree of 0 is ?1. OPERATIONS ON REAL ALGEBRAIC NUMBERS Note that if and are real algebraic numbers, then so are ? , ?1 ( assuming 6 = 0), + , and .  ... 
doi:10.1201/9781420035315.ch33 fatcat:on3snkonknhxpmruozfldyyvg4

A Verified Implementation of Algebraic Numbers in Isabelle/HOL

Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada
2018 Journal of automated reasoning  
We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers.  ...  We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution  ... 
doi:10.1007/s10817-018-09504-w pmid:32226180 pmcid:PMC7089722 fatcat:pvzb5tg36jdb5bfrvnlgmajq7y

Page 2164 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews  
The group of the order 2, generated by complex conjugation, acts on the set X¥(C) of complex points of any real algebraic variety X.  ...  , one can ask whether or not C has an infinite number of real points, i.e., if the trace of C in R? is a real curve.  ... 

Complexity of Symbolic and Numerical Problems (Dagstuhl Seminar 15242)

Peter Bürgisser, Felipe Cucker, Marek Karpinski, Nicolai Vorobjov, Marc Herbstritt
2016 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 15242 "Complexity of Symbolic and Numerical Problems".  ...  The authors' interest for these problems was sparked by connections between lower bounds in algebraic complexity theory and upper bounds on the number of real roots of "sparse like" polynomials. probability  ...  In earlier work we could prove the real number PCP theorem to hold along the lines of Dinur's proof. In this talk we report on an algebraic proof of the theorem.  ... 
doi:10.4230/dagrep.5.6.28 dblp:journals/dagstuhl-reports/BurgisserCKV15 fatcat:xje5zmbik5cv5ebacrqhmmgz5i

Computer Arithmetic of Numbers, Vectors, Figures and Functions. Algorithms and Hardware

Solomon Khmelnik
2020 Zenodo  
The article contains a prospectus of the book under the same title [1]. This book is published only in Russian and in this connection, this prospectus is published. The book contains 673 pages.  ...  The author seeks assistance in publishing a book in English.  ...  Computer arithmetic of complex mathematical objects originates in the article by Shannon on the positional coding of real numbers on a negative basis [1].  ... 
doi:10.5281/zenodo.3920212 fatcat:2d4ih7xtunb4dcrq6zjjmm43y4

Computations with one and two real algebraic numbers [article]

Ioannis Z. Emiris, Elias P. Tsigaridas
2005 arXiv   pre-print
We present algorithmic and complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer  ...  Our main results, in the univariate case, concern the problems of real root isolation (Th. 19) and simultaneous inequalities (Cor.26) and in the bivariate, the problems of system real solving (Th.42),  ...  University of Athens, and IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-006413-2 (ACS -Algorithms for Complex Shapes).  ... 
arXiv:cs/0512072v1 fatcat:liaima3txnfivml4hmke3c4gsi

Page 14 of Mathematical Reviews Vol. , Issue 93f [page]

1993 Mathematical Reviews  
It is known that algebraic function fields in one variable over the field C of complex-numbers or, equivalently, complex irreducible algebraic curves, in geometrical terms, are nothing else but compact  ...  This paper surveys the theoretical computational aspects of real algebraic geometry, for the benefit of computational geometers unfamiliar with real algebraic geometry.  ... 

Computing Singular Points of Projective Plane Algebraic Curves by Homotopy Continuation Methods

Zhongxuan Luo, Erbao Feng, Jielin Zhang
2014 Discrete Dynamics in Nature and Society  
We prove that the algorithm has the polynomial time complexity on the degree of the algebraic curve.  ...  We present an algorithm that computes the singular points of projective plane algebraic curves and determines their multiplicities and characters. The feasibility of the algorithm is analyzed.  ...  We also outline the algorithm on computing the singular points of projective plane algebraic curves, and afterwards we analyze feasibility and complexity of the algorithm.  ... 
doi:10.1155/2014/230847 fatcat:nmu5oh5vgbdgxf6cqsvouphxq4

Page 101 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
NY) Computing the irreducible real factors and components of an algebraic curve.  ...  Our construction is based on computing the ir- reducible complex factors and then investigating high precision  ... 

On Computing a Set of Points Meeting Every Cell Defined by a Family of Polynomials on a Variety

Saugata Basu, Richard Pollack, Marie-Françoise Roy
1997 Journal of Complexity  
The number of semi-algebraically connected components of all non-empty sign conditions on P over V is bounded by s (O(d)) .  ...  In this paper we present a new algorithm to compute a set of points meeting every semi-algebraically connected component of each non-empty sign condition of P over V . Its complexity is s d .  ...  The bound in Theorem 1 on the number of cells of on the variety is separated into a combinatorial part (the dependence on ) which depends only on the real dimension of the variety and an algebraic part  ... 
doi:10.1006/jcom.1997.0434 fatcat:pexulakn4bcylhdaruqmg3iryu

Page 1991 of Mathematical Reviews Vol. , Issue 84e [page]

1984 Mathematical Reviews  
The computational complexity of a real number is approximately the complexity of recognizing prefixes of its binary representation.  ...  For earlier work on complexity and real numbers see the paper by the author and H. Friedman [Theoret. Comput. Sci. 20 (1982), 323-352; MR 83j:03103].  ... 

Page 7695 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
(I-PISA; Pisa) Algorithms to compute the topology of orientable real algebraic surfaces. (English summary) International Symposium on Symbolic and Algebraic Computation (ISSAC’2002) (Lille). J.  ...  work on Eremenko and Gabrielov giving lower a priori bounds for the number of real solutions to certain enumerative problems.  ... 

Page 8591 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
In the present paper, this bound is improved for real algebraic curves. Theorem. Let C be a real algebraic curve and s be the number of real branches.  ...  Mayer-Vietoris double complex of the arrangement degenerates in one step.  ... 
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