Nash Functions
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In real algebraic geometry, a Nash function on an open semialgebraic subset U of Rn is an analytic function f: U - R satisfying a non trivial 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.) Polynomial and regular rational functions are Nash functions; the functio...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In real algebraic geometry, a Nash function on an open semialgebraic subset U of Rn is an analytic function f: U - R satisfying a non trivial 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.) Polynomial and regular rational functions are Nash functions; the function xmapsto 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. Actually, Nash functions are those functions needed in order to have an implicit function theorem in real algebraic geometry.
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