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… 
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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.
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.
...
...