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Approximation of $C^\infty$-functions without changing their zero-set
For a C ∞ function φ:M→ℝ (where M is a real algebraic manifold) the following problem is studied. If φ -1 (0) is an algebraic subvariety of M, can φ be approximated by rational regular functions f such that f -1 (0)=φ -1 (0)?