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It is not trivial to prove that this procedure will always terminate after finitely many such extensions of the basis.


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A collection of multivariate polynomials defines the geometric object in the multidimensional space consisting of those points where all the polynomials in the collection vanish simultaneously. The other relevant points can be described implicitly by equations that must be satisfied by their coordinates. In this simple example, all colorings are identical up to permutation of colors. Many analytic functions can be defined by some finite set of initial values and a finite set of differential equations they satisfy.

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Also recurrence equations for infinite sums can be found by a similar technique. The key idea is to encode the number of coins as exponents of variables.

One of the aims of coding theory is to conceive schemes for transmitting information through a communication channel in such a way that errors that may possibly occur during the transmission can be detected and corrected by the receiver. If not, i. This solves the error correction problem in principle. However, the method given here is a somewhat oversimplified version of what is actually applied in practice.

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In order to obtain efficient error correcting algorithms, refined techniques have to be applied which are beyond the scope of this text. In particular, the appropriate coefficient fields in the context of coding are finite and specific algorithmic techniques available in such fields can be applied. The "general inverse kinematics problem" in robotics asks for determining how the various joints in a robot have to be turned in order to reach a certain position.

For example, given a point in 3D space and an orientation, we may ask for the appropriate angles of the joints in order to reach the point with the requested orientation. This proves the correctness of the program.

The oil compartments in a field are not independent from each other but there are geological interactions between them, so that it is possible that reducing the extraction rate at one access point may increase the total production of all points. These interactions are hard to model as they depend on parameters which are difficult to measure. A pragmatic approach is therefore to directly establish the relations between extraction parameters at the access points and the resulting production gains.

We encourage experts in the various application fields to complement this article by additional subsections describing these applications.

Cylindrical algebraic decomposition

Bruno Buchberger and Manuel Kauers , Scholarpedia, 5 10 Jump to: navigation , search. Post-publication activity Curator: Manuel Kauers Contributors:. Riccardo Guida. Willy Hereman. Sponsored by: Eugene M. Izhikevich , Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia Sponsored by: Prof.

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Willy Hereman , Dept. Finally I decided that I should really take the time to write the material up in a coherent way and make a book out of it. Here, now, is the result of this work.


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Over the years many students have been helpful in improving the quality of the notes, and also several colleagues at Linz and elsewhere have contributed to it. I want to thank them all for their effort, in particular I want to thank B.


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Buchberger, who taught me the theory of Grabner bases nearly two decades ago, B. Caviness and B.

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Saunders, who first stimulated my interest in various problems in computer algebra, G. Collins, who showed me how to compute in algebraic domains, and J. Sendra, with whom I started to apply computer algebra methods to problems in algebraic geometry. Several colleagues have suggested improvements in earlier versions of this book. However, I want to make it clear that I am responsible for all remaining mistakes.

JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Mathematics Algebra. Buy eBook. Buy Softcover. FAQ Policy. About this Textbook For several years now I have been teaching courses in computer algebra at the Universitat Linz, the University of Delaware, and the Universidad de Alcala de Henares.