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Complexities Approach to Two Problems In Number Theory
[article]

2016
*
arXiv
*
pre-print

By Kolmogorov

arXiv:1610.04026v3
fatcat:5udl5eurnzh63c6ysh6b7kta6y
*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*. ...##
###
Algorithms in Real Algebraic Geometry: A Survey
[article]

2014
*
arXiv
*
pre-print

We also describe some recent results linking

arXiv:1409.1534v1
fatcat:nyprfglktvdtnmhu3zwqrb547y
*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. ...##
###
Computational Real Algebraic Geometry
[chapter]

2004
*
Handbook of Discrete and Computational Geometry, Second Edition
*

1) 2 : If,

doi:10.1201/9781420035315.ch33
fatcat:on3snkonknhxpmruozfldyyvg4
*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 . ...##
###
A Verified Implementation of Algebraic Numbers in Isabelle/HOL

2018
*
Journal of automated reasoning
*

We formalize

doi:10.1007/s10817-018-09504-w
pmid:32226180
pmcid:PMC7089722
fatcat:pvzb5tg36jdb5bfrvnlgmajq7y
*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 ...##
###
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)

2016
*
Dagstuhl Reports
*

This report documents

doi:10.4230/dagrep.5.6.28
dblp:journals/dagstuhl-reports/BurgisserCKV15
fatcat:xje5zmbik5cv5ebacrqhmmgz5i
*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. ...##
###
Computer Arithmetic of Numbers, Vectors, Figures and Functions. Algorithms and Hardware

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]. ...

##
###
Computations with one and two real algebraic numbers
[article]

2005
*
arXiv
*
pre-print

We present algorithmic and

arXiv:cs/0512072v1
fatcat:liaima3txnfivml4hmke3c4gsi
*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). ...##
###
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

2014
*
Discrete Dynamics in Nature and Society
*

We prove that

doi:10.1155/2014/230847
fatcat:nmu5oh5vgbdgxf6cqsvouphxq4
*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. ...##
###
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

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 ...

##
###
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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