Lesson plan
Nash function
In real algebraic geometry, a Nash function on an open semialgebraic subset U ⊂ R n 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 R n is a subset obtained from subsets of the form {x in R n : P(x)=0} or {x in R n : 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. * 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.
Article summary
In real algebraic geometry, a Nash function on an open semialgebraic subset U ⊂ R n 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 R n is a subset obtained from subsets of the form {x in R n : P(x)=0} or {x in R n : 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. * 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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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