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 definable in terms of polynomial identities and inequalities. The theorem—also known as the Tarski–Seidenberg projection property—is named after Alfred Tarski and Abraham Seidenberg. It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations and inequalities by logical connectors ∨ (or), ∧ (and), ¬ (not) and quantifiers ∀ (for all), ∃ (exists) is equivalent with a similar formula without quantifiers. An important consequence is the decidability of the theory of real-closed fields.

Although the original proof of the theorem was constructive, the resulting algorithm has a computational complexity that is too high for using the method on a computer. George E. Collins introduced the algorithm of cylindrical algebraic decomposition, which allows quantifier elimination over the reals in double exponential time. This complexity is optimal, as there are examples where the output has a double exponential number of connected components. This algorithm is therefore fundamental, and it is widely used in computational algebraic geometry.

Statement

A semialgebraic set in Rn is a finite union of sets defined by a finite number of polynomial equations and inequalities, that is by a finite number of statements of the form

and

for polynomials p and q. We define a projection map π : Rn+1 → Rn by sending a point (x1,...,xn,xn+1) to (x1,...,xn). Then the Tarski–Seidenberg theorem states that if X is a semialgebraic set in Rn+1 for some n > 1, then π(X) is a semialgebraic set in Rn.

Failure with algebraic sets

If we only define sets using polynomial equations and not inequalities then we define algebraic sets rather than semialgebraic sets. For these sets the theorem fails, i.e. projections of algebraic sets need not be algebraic. As a simple example consider the circle in R2 defined by the equation

This is a perfectly good algebraic set, but project it down by sending (x,y) in R2 to x in R and we have the set of points satisfying -1 ≤ x ≤ 1. This is a semialgebraic set as we would expect from the theorem, but it is not an algebraic set.

Relation to structures

This result confirmed that semialgebraic sets in Rn form what is now known as an o-minimal structure on R. These are collections of subsets Sn of Rn for each n ≥ 1 such that we can take finite unions and complements of the subsets in Sn and the result will still be in Sn, moreover the elements of S1 are simply finite unions of intervals and points. The final condition for such a collection to be an o-minimal structure is that the projection map on the first n coordinates from Rn+1 to Rn must send subsets in Sn+1 to subsets in Sn. The Tarski–Seidenberg theorem tells us this holds if Sn is the set of semialgebraic sets in Rn.

See also

Tarski–Seidenberg theorem at PlanetMath.org

References

  • van den Dries, L. (1998). Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series. 248. Cambridge University Press. Zbl 0953.03045.
  • Khovanskii, A.G. (1991). Fewnomials. Translations of Mathematical Monographs. 88. Translated from the Russian by Smilka Zdravkovska. Providence, RI: American Mathematical Society. ISBN 0-8218-4547-0. Zbl 0728.12002.
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