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### Bounding the number of connected components of a real algebraic set

Riccardo Benedetti, Francois Loeser, Jean Jacques Risler
1991 Discrete & Computational Geometry
For every polynomial map f = (fi ..... fk): ~"~ Rk, we consider 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

Sal Barone, Saugata Basu
2011 Discrete & Computational Geometry
We prove that 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]

Sal Barone, Saugata Basu
2011 arXiv   pre-print
We prove that 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]

Saugata Basu
2014 arXiv   pre-print
We survey both old and new developments in 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

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 .  ...  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]

Saugata Basu, Richard Pollack, Marie-Francoise Roy
2009 arXiv   pre-print
We prove an asymptotically tight 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]

Saugata Basu, Marie-Francoise Roy
2009 arXiv   pre-print
We prove explicit 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

Saugata Basu, Richard Pollack, Marie-Françoise Roy
2009 Combinatorica
We prove an asymptotically tight 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

Huu Phuoc Le, Mohab Safey El Din, Timo de Wolff
2020 Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation
Let be 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

S. Basu, M.-F. Roy, M. Safey El Din, É. Schost
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]

Bhubaneswar Mishra
2004 Handbook of Discrete and Computational Geometry, Second Edition
j : C i = j C j : Computational 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]

J. Maurice Rojas
2000 arXiv   pre-print
We 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  ...  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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