mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, cylindrical algebraic decomposition (CAD) is a notion, and an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

to compute it, that are fundamental for

computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...

and real algebraic geometry. Given a set ''S'' of polynomials in R^''n'', a cylindrical algebraic decomposition is a decomposition of R^''n'' into connected

semialgebraic set In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and ine ...

s called ''cells'', on which each polynomial has constant sign, either +, − or 0. To be ''cylindrical'', this decomposition must satisfy the following condition: If 1 ≤ ''k'' < ''n'' and ''π'' is the projection from R^''n'' onto R^{''n''−''k''} consisting in removing the last ''k'' coordinates, then for every pair of cells ''c'' and ''d'', one has either ''π''(''c'') = ''π''(''d'') or ''π''(''c'') ∩ ''π''(''d'') = ∅. This implies that the images by ''π'' of the cells define a cylindrical decomposition of R^{''n''−''k''}. The notion was introduced by George E. Collins in 1975, together with an

for computing it. Collins' algorithm has a

computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...

that is double exponential in ''n''. This is an upper bound, which is reached on most entries. There are also examples for which the minimal number of cells is doubly exponential, showing that every general algorithm for cylindrical algebraic decomposition has a double exponential complexity. CAD provides an effective version of quantifier elimination over the reals that has a much better computational complexity than that resulting from the original proof of

Tarski–Seidenberg theorem In mathematics, the Tarski–Seidenberg theorem states that a set in (''n'' + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto ''n''-dimensional space, and the resulting set is still defi ...

. It is efficient enough to be implemented on a computer. It is one of the most important algorithms of computational real algebraic geometry. Searching to improve Collins' algorithm, or to provide algorithms that have a better complexity for subproblems of general interest, is an active field of research.

Implementations

Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiza ...

CylindricalDecomposition

* ttp://redlog.eu/ redlog* Maple
The RegularChains Library
an

References

*Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10. Springer-Verlag, Berlin, 2006. x+662 pp. ; 3-540-33098-4 *Strzebonski, Adam.

' from

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...

.
Cylindrical Algebraic Decomposition
in ''Planning algorithms'' by Steven M. LaValle. Accessed 13 July 2007 *Caviness, Bob; Johnson, Jeremy
Quantifier Elimination and Cylindrical Algebraic Decomposition
Texts and Monographs in Symbolic Computation. Springer-Verlag, Berlin, 1998. *Collins, George E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, Second GI Conf. Automata Theory and Formal Languages, Springer LNCS 33, 1975. *Davenport, James H.; Heintz, Joos: Real quantifier elimination is doubly exponential, Journal of Symbolic Computation, 1988. Volume 5, Issues 1–2, ISSN 0747-7171, Algebra Real algebraic geometry {{algebraic-geometry-stub