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Bounding the number of connected components of a real algebraic set

1991
*
Discrete & Computational Geometry
*

For every polynomial map f = (fi ..... fk): ~"~ Rk, we consider

doi:10.1007/bf02574685
fatcat:go6mctzxpjacpppmyoyul6lt2i
*the**number**of**connected**components**of*its zero*set*, B(Zf), and two natural "measures*of**the*complexity*of*f," that is*the*triple (n, k, d ...*The*two*bounds*are obtained by*a*similar technique involving*a*slight modification*of*Milnor-Thom's argument, Smith's theory, and information about*the*sum*of*Betti*numbers**of*complex complete intersections ...*Bounding**the**Number**of**Connected**Components**of**a**Real**Algebraic**Set*193 2. ...##
###
Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

2011
*
Discrete & Computational Geometry
*

We prove that

doi:10.1007/s00454-011-9391-3
fatcat:pf2pms36t5fzjgioei7m5kvsmu
*the**number**of*semi-*algebraically**connected**components**of**the*realizations*of*all realizable sign conditions*of**the*family P on V is*bounded*by where s = card P, and X 1≤j≤k " s j " 4 j d( ... Let V ⊂ R k be*the**real**algebraic*variety defined by*the*polynomials in Q and suppose that*the**real*dimension*of*V is*bounded*by k . ... Roy for making helpful comments on*a*first draft*of**the*paper.*The*authors were partially supported by an NSF grant CCF-0915954. ...##
###
Refined bounds on the number of connected components of sign conditions on a variety
[article]

2011
*
arXiv
*
pre-print

We prove that

arXiv:1104.0636v4
fatcat:yggsqd25gffghfvxrltiuqiek4
*the**number**of*semi-*algebraically**connected**components**of**the*realizations*of*all realizable sign conditions*of**the*family P on V is*bounded*by ∑_j=0^k'4^js +1 jF_d,d_0,k,k'(j), where s = ... Let V ⊂^k be*the**real**algebraic*variety defined by*the*polynomials in Q and suppose that*the**real*dimension*of*V is*bounded*by k'. ... Roy for making helpful comments on*a*first draft*of**the*paper.*The*authors were partially supported by an NSF grant CCF-0915954. ...##
###
Algorithms in Real Algebraic Geometry: A Survey
[article]

2014
*
arXiv
*
pre-print

We survey both old and new developments in

arXiv:1409.1534v1
fatcat:nyprfglktvdtnmhu3zwqrb547y
*the*theory*of*algorithms in*real**algebraic*geometry -- starting from effective quantifier elimination in*the*first order theory*of**reals*due to Tarski and Seidenberg ... 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*... Counting*the**number**of*semi-*algebraically**connected**components**of*such*sets*is even harder. ...##
###
Page 697 of Mathematical Reviews Vol. , Issue 94b
[page]

1994
*
Mathematical Reviews
*

This theorem gives

*a**bound*on*the**number**of**connected**components**of*such*a*variety as*a*function*of*nm and*the*degree*of**the*variety. ... In several applications*of*this result,*the*inequality has been reversed in such*a*way that*the*upper*bound*for*the**number**of**connected**components**of*some*algebraic**sets*translates to lower*bounds*for ...##
###
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 . ...

*A*semi-

*algebraic*

*set*has

*a*finite

*number*

*of*semialgebraically

*connected*

*components*, each

*of*which is

*a*semi-

*algebraic*

*set*. ...

##
###
An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions
[article]

2009
*
arXiv
*
pre-print

We prove an asymptotically tight

arXiv:math/0603256v3
fatcat:n65ys7z3z5grhi7axqgj4tpury
*bound*(asymptotic with respect to*the**number**of*polynomials for fixed degrees and*number**of*variables) on*the**number**of*semi-*algebraically**connected**components**of**the*realizations ... More precisely, we prove that*the**number**of*semi-*algebraically**connected**components**of**the*realizations*of*all realizable sign conditions*of**a*family*of*s polynomials in [X_1,... ... Thus, it suffices to prove*the*lemma in case C is*a*semi-*algebraically**connected**component**of**a**real**algebraic**set*. ...##
###
Bounding the radii of balls meeting every connected component of semi-algebraic sets
[article]

2009
*
arXiv
*
pre-print

We prove explicit

arXiv:0911.1340v1
fatcat:gievc3lrunfy5mi7bobe33kqte
*bounds*on*the*radius*of**a*ball centered at*the*origin which is guaranteed to contain all*bounded**connected**components**of**a*semi-*algebraic**set*S ⊂R^k defined by*a*quantifier-free formula ... We also prove*a*similar*bound*on*the*radius*of**a*ball guaranteed to intersect every*connected**component**of*S (including*the*unbounded*components*). ... Proof*of*Theorem 4: Since every semi-*algebraically**connected**component**of**the*realization*of**a*weak sign condition on P must contain*a**connected**component**of*some*algebraic**set*Zer(P ′ , R k ), where P ...##
###
Page 5235 of Mathematical Reviews Vol. , Issue 2004g
[page]

2004
*
Mathematical Reviews
*

*connected*

*components*

*of*

*algebraic*

*sets*. ... As another generalization

*of*Descartes’ result to higher di- mensions, similar sharpenings for

*the*

*number*

*of*compact and non-compact

*connected*

*components*

*of*

*the*zero

*set*in R”

*of*

*a*single sparse polynomial ...

##
###
An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions

2009
*
Combinatorica
*

We prove an asymptotically tight

doi:10.1007/s00493-009-2357-x
fatcat:2hbq2vghfrf7hd7pudhx376p74
*bound*(asymptotic with respect to*the**number**of*polynomials for fixed degrees and*number**of*variables) on*the**number**of*semi-*algebraically**connected**components**of**the*realizations ... More precisely, we prove that*the**number**of*semi-*algebraically**connected**components**of**the*realizations*of*all realizable sign conditions*of**a*family*of*s polynomials in R[X 1 , . . . , X k ] whose degrees ... Thus, it suffices to prove*the*lemma in case C is*a*semi-*algebraically**connected**component**of**a**real**algebraic**set*. ...##
###
Computing the real isolated points of an algebraic hypersurface

2020
*
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation
*

Let be

doi:10.1145/3373207.3404049
dblp:conf/issac/LeDW20
fatcat:hyhczrhu3rbu5h64y44ggqcste
*the*field*of**real**numbers*. We consider*the*problem*of*computing*the**real*isolated points*of**a**real**algebraic**set*in given as*the*vanishing*set**of**a*polynomial system. ... It is based on*the*computations*of*critical points as well as roadmaps for answering*connectivity*queries in*real**algebraic**sets*. ... Recall that*real**algebraic**sets*have*a*finite*number**of*semi-*algebraically**connected**components*[2, Theorem 5.21]. Let be*a*semi-*algebraically**connected**component**of*ℋ ∩ . ...##
###
A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets

2014
*
Foundations of Computational Mathematics
*

*components*

*of*

*a*

*real*

*algebraic*

*set*, Zer(Q, R k ), whose complexity is also

*bounded*by

*The*best previously known algorithm for constructing

*a*roadmap

*of*

*a*

*real*

*algebraic*subset

*of*R k defined by

*a*polynomial ...

*The*complexity

*of*

*the*algorithm, measured by

*the*

*number*

*of*arithmetic operations in

*the*domain D, is

*bounded*by As

*a*consequence, there exist algorithms for computing

*the*

*number*

*of*semi-

*algebraically*

*connected*...

*of*semi-

*algebraically*

*connected*

*components*

*of*

*a*given semi-

*algebraic*

*set*S ⊂ R k where R is

*a*

*real*closed field (for example

*the*field

*of*

*real*

*numbers*), is

*a*very important problem in algorithmic semi-

*algebraic*...

##
###
Computational Real Algebraic Geometry
[chapter]

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

j : C i = j C j : Computational

doi:10.1201/9781420035315.ch33
fatcat:on3snkonknhxpmruozfldyyvg4
*Real**Algebraic*Geometry 7*CONNECTED**COMPONENTS**OF*SEMI-*ALGEBRAIC**SETS**A*consequence*of**the*Milnor-Thom result Mil64, Tho65] gives*a**bound*for*the**number*(zeroth Betti ... GLOSSARY*connected**component**of**a*semi-*algebraic**set*:*A*maximal*connected*subset*of**a*semi-*algebraic**set*. ...##
###
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
[article]

2000
*
arXiv
*
pre-print

We give new positive and negative results (some conditional) on speeding up computational

arXiv:math/9905004v2
fatcat:2oemu2szkbabldcbhhsj6a4isu
*algebraic*geometry over*the**reals*: (1)*A*new and sharper upper*bound*on*the**number**of**connected**components**of**a*... 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. ... Finding an optimal upper*bound*on*the**number**of**connected**components**of**a*semi-*algebraic**set*, even in*the*special case*of*nondegenerate*real**algebraic**sets*, remains an open problem. ...##
###
Page 5384 of Mathematical Reviews Vol. , Issue 2000h
[page]

2000
*
Mathematical Reviews
*

It is well-known that S has

*a*finite*number**of*semi-*algebraically**connected**components*. ...*The*space*of*non-singular*real**algebraic*curves*of*bidegree (4,3) on*a*hyperboloid consists*of**a*finite*number**of**connected*com- ponents. ...
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