Multiplicative calculus

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.

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

$p(x_1,\dots ,x_n)=0$

and

$q(x_1,\dots ,x_n)>0$

for polynomials p and q. We define a projection map π : Rⁿ⁺¹ → Rⁿ by sending a point (x₁,...,x_n,x_n+1) to (x₁,...,x_n). 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. As a simple example consider the circle in R² defined by the equation

$x^2+y^2-1=0.$

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 S_n of Rⁿ for each n ≥ 1 such that we can take finite unions and complements of the subsets in S_n and the result will still be in S_n, 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 S_n+1 to subsets in S_n. The Tarski–Seidenberg theorem tells us this holds if S_n is the set of semialgebraic sets in Rⁿ.

References

• L. van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series. 248, Cambridge University Press 1998.