Nash functions

In real algebraic geometry, a Nash function on an open semialgebraic subset U ⊂ Rⁿ is an analytic function f: U → R satisfying a nontrivial polynomial equation P(x,f(x)) = 0 for all x in U (A semialgebraic subset of Rⁿ is a subset obtained from subsets of the form {x in Rⁿ : P(x)=0} or {x in Rⁿ : P(x) > 0}, where P is a polynomial, by taking finite unions, finite intersections and complements). Some examples of Nash functions:

• Polynomial and regular rational functions are Nash functions.
• ${\displaystyle x\mapsto {\sqrt {1+x^{2}}}}$ is Nash on R.
• the function which associates to a real symmetric matrix its i-th eigenvalue (in increasing order) is Nash on the open subset of symmetric matrices with no multiple eigenvalue.

Nash functions are those functions needed in order to have an implicit function theorem in real algebraic geometry.