Computational Real Algebraic Geometry

@inproceedings{Mishra2004ComputationalRA,
  title={Computational Real Algebraic Geometry},
  author={Bud Mishra},
  booktitle={Handbook of Discrete and Computational Geometry, 2nd Ed.},
  year={2004}
}
  • B. Mishra
  • Published in
    Handbook of Discrete and…
     2004
  • Mathematics, Computer Science
Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad and general class of problems arising in robotics, vision, computer aided design, geometric theorem proving, etc. The following survey paper discusses the underlying concepts, algorithms and a series… 
View via Publisher
60 Citations
Background Citations
18
Methods Citations
6

60 Citations

Citation Type

Citation Type

Geometric algorithms for algebraic curves and surfaces
TLDR
This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree and proposes a new method to compute an isotopic triangulation of an algebraic surface, which is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.
Some Methods of Problem Solving in Elementary Geometry
  • T. Hales
  • Mathematics
    22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
  • 2007
TLDR
The methods that were used in the original proof of the Kepler conjecture are investigated and a number of other methods that might be used to automate the proofs of these problems are described.
LINE PROBLEMS IN NONLINEAR COMPUTATIONAL GEOMETRY
We first review some topics in the classical computational geometry of lines, in particular the O(n 3+ǫ ) bounds for the combinatorial complexity of the set of lines in R 3 interacting with n objects…
An exact and efficient approach for computing a cell in an arrangement of quadrics
Lecture 0: Introduction
This lecture is an orientation on the central problems that concern us. Specifically, we identify three families of " Fundamental Problems " in algorithmic algebra (§1 – §3). In the rest of the…
Subquadratic Algorithms for Algebraic Generalizations of 3SUM
TLDR
It is proved that there exist bounded-degree algebraic decision trees of depth $O (n^{\frac{12}{7}+\varepsilon})$ that solve 3POL, and that 3POL can be solved in $O(n^2 {(\log \log n)}^\frac{3}{2} /{(\log n)$ time in the real-RAM model.
Computing Smooth Feedback Plans Over Cylindrical Algebraic Decompositions
TLDR
This paper builds a vector field over the decomposition of cylindrical algebraic decompositions, which implies that it can obtain smooth feedback plans for the generalized piano movers’ problem.
Computing at Least One of Two Roots of a Polynomial Is, in General, Not Algorithmic
TLDR
The proof shows that such an algorithm is not possible even for computable polynomial equations, which means that in general, it is easier to find a point (x1, xn), in which a given system of equations has a unique solution than when this system has several solutions.
Simple and Efficient Algorithms for Computing Smooth , Collision-Free Feedback Laws
TLDR
A novel approach to computing feedback laws in the presence of obstacles by computing a vector field over the entire state space; all trajectories obtained from following this vector field are guaranteed to asymptotically reach the goal state.
Constraint Problems: Computability Is Equivalent to Continuity
TLDR
This paper provides an algorithm that computes a set of all possible values that satisfy given constraints for all the cases when the dependence of the desired set on the parameters of the problem is not continuous.
...
...

References

SHOWING 1-10 OF 32 REFERENCES
Some algebraic and geometric computations in PSPACE
TLDR
A PSPACE algorithm for determining the signs of multivariate polynomials at the common zeros of a system of polynomial equations is given and it is shown that the existential theory of the real numbers can be decided in PSPACE.
On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds
On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals
  • J. Renegar
  • Mathematics, Computer Science
    J. Symb. Comput.
  • 1992
Algorithms in real algebraic geometry
Solving Systems of Polynomial Inequalities in Subexponential Time
Geometric and Solid Modeling: An Introduction
TLDR
"Geometric and Solid Modeling" deals with the concepts and tools needed to design and implement solid-modeling systems and their infrastructure and substrata, making this information remarkably accessible--to the novice as well as to the experienced designer.
Computational geometry: a retrospective
TLDR
This work will survey some of its principal accomplishments, and in light of recent developments, it will discuss the profound transformations the field has begun to undergo.
The complexity of robot motion planning
TLDR
John Canny resolves long-standing problems concerning the complexity of motion planning and, for the central problem of finding a collision free path for a jointed robot in the presence of obstacles, obtains exponential speedups over existing algorithms by applying high-powered new mathematical techniques.
A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials
TLDR
A new algorithm is presented which computes a point in each connected component of each non-empty sign condition over P1,…,P s which is nearly optimal in the sense that the size of the output can be as large as s(O(sd/k) k .
Improved Algorithms for Sign Determination and Existential Quantifier Elimination
TLDR
A new sign determination method based on the earlier algorithm, but with two advantages: it is faster in the univariate case, and it allows purely symbolic quantifier elimination in pseudo-polynomial time.
...
...