Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
Cylindrical algebraic decomposition II: an adjacency algorithm for the plane
SIAM Journal on Computing
The complexity of elementary algebra and geometry
Journal of Computer and System Sciences
A cluster-based cylindrical algebraic decomposition algorithm
Journal of Symbolic Computation
On the sign of a real algebraic number
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
A cluster-based cylindrical algebraic decomposition algorithm
Journal of Symbolic Computation
A bibliography of quantifier elimination for real closed fields
Journal of Symbolic Computation
Towards computing non algebraic cylindrical decompositions
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Working with real algebraic plane curves in REDUCE the GCUR package
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Examples of automatic theorem proving a real geometry
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
Numeric-symbolic algorithms for evaluating one-dimensional algebraic sets
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
Representing 2D Digital Objects
DGCI '00 Proceedings of the 9th International Conference on Discrete Geometry for Computer Imagery
Computing the topology of real algebraic surfaces
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Computer algebra handbook
Algorithms to compute the topology of orientable real algebraic surfaces
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
On the exact computation of the topology of real algebraic curves
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
An exact and efficient approach for computing a cell in an arrangement of quadrics
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Complete subdivision algorithms, I: intersection of Bezier curves
Proceedings of the twenty-second annual symposium on Computational geometry
A delineability-based method for computing critical sets of algebraic surfaces
Journal of Symbolic Computation
Computing the topology of a real algebraic plane curve whose equation is not directly available
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On the complexity of real solving bivariate systems
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Journal of Computational and Applied Mathematics
Global minimization of rational functions and the nearest GCDs
Journal of Global Optimization
Topology of real algebraic space curves
Journal of Symbolic Computation
On the computation of the topology of a non-reduced implicit space curve
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
ACM Transactions on Graphics (TOG)
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
On the topology of planar algebraic curves
Proceedings of the twenty-fifth annual symposium on Computational geometry
Computing nearest Gcd with certification
Proceedings of the 2009 conference on Symbolic numeric computation
An exact and efficient approach for computing a cell in an arrangement of quadrics
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Computation of the topology of real algebraic space curves
Journal of Symbolic Computation
Topology of 2D and 3D rational curves
Computer Aided Geometric Design
A subdivision method for computing nearest gcd with certification
Theoretical Computer Science
A worst-case bound for topology computation of algebraic curves
Journal of Symbolic Computation
Determining the topology of real algebraic surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
On the isotopic meshing of an algebraic implicit surface
Journal of Symbolic Computation
Journal of Symbolic Computation
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It was proved over a century ago that an algebraic curve C in the real projective plane, of degree n, has at most (n-1)(n-2)2+1 connected components. If C is nonslngular, then each of its commponents is a topological circle. A circle in the projectlve plane either separates it into a disk (the interior of the circle) and a Mobius band (the circle's exterior), or does not separate it. In the former case, the circle is an oval. If C is nonsingular, then all its components are ovals if n is even, and all except one are ovals if n is odd. An oval is included in another if it lies in the other's interior. The topological type of (a nonsingular) C is completely determined by (1) the parity of n, (2) how many ovals it has, and (3) the partial ordering of its ovals by inclusion. We present an algorithm which, given a homogeneous polinomial f(x,y,z) of degree n with integer coefficients, checks whether tlte curve defined hy f = 0 is nonsingular and if so, computes its topological type. The algorithm's maximum computing time is O(n^2^7L(d)^3), where d is the sum of the absolute values of the integer coofficients of f, and L(d) is the length of d.