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Mathematica' CylindricalDecomposition implements an algorithm known as Cylindrical Algebraic Decomposition. Wolfram MathWorld's article on Cylindrical Algebraic Decomposition says that this algorithm "becomes computationally infeasible for complicated inequalities."

Can this statement be made more precise? Specifically, how does the time and space relate to the degree and number of variables of the multivariate polynomials? Does the time and space depend on other parameters?

Dr. belisarius
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Tyson Williams
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1 Answers1

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Tarski showed that for every formula including quantifiers there is always an equivalent quantifier free formula. Obtaining the latter from the former is called quantifier elimination. (...)

In particular, for the cylindrical algebraic decomposition (CAD), the number of operations usually scales in a doubly exponential fashion with the number of variables, while the newer methods are doubly exponential in the number of quantifier alternations.

Reference: MIT 6.972 Algebraic techniques and semidefinite optimization by Pablo A. Parrilo

Edit: A nice article on Mma CAD algorithms here

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Dr. belisarius
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    Tarski and his paradox are what convinced me that I was right in not becoming a mathematician after all... Funny meeting him here! – acl Jun 20 '11 at 01:39
  • @belisarius Thanks! Do you know the dependence on the degree of the polynomials as well? – Tyson Williams Jun 20 '11 at 02:22
  • @Tyson Nope, I was involved in QE a long time ago, and my memory fails. If it is _really_ critical for you to know, I may try unpacking old notes archival boxes in the cellar. But that shall take a while, and I'm not really sure whether they are there. – Dr. belisarius Jun 20 '11 at 02:41
  • @belisarius No, you don't have to do that. I am already intrigued by the newer algorithms mentioned in your reference that are _only_ doubly exponential in the number of quantifier alternations since I have a problem with at most five variables and no quantifiers. Do you know if Mathematica implements any of these newer algorithms? – Tyson Williams Jun 20 '11 at 13:12
  • @Tyson I think I've read somewhere they used to use Collin's algorithm, but of course that should be confirmed by WRI personnel. – Dr. belisarius Jun 20 '11 at 17:46
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    @Tyson Found an article http://reference.wolfram.com/mathematica/tutorial/RealPolynomialSystems.html – Dr. belisarius Jun 20 '11 at 17:50
  • @belisarius Yes, Wolfram MathWorld's [Tarski's theorem][http://mathworld.wolfram.com/TarskisTheorem.html] confirms that Collin's algorithm is Mathematica' CylindricalDecomposition. Your link is very good. It says that CylindricalDecomposition is the main workhorse and is assisted by several other special case algorithms. However, my problem does not fit into any of the special cases :( – Tyson Williams Jun 20 '11 at 19:26
  • I had never heard of these functions before. what do you do @belisarius ? are you a mathematician? –  Jun 21 '11 at 01:53