Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry
It is known that a p-adic, locally Lipschitz continuous semi-algebraic function, is piecewise Lipschitz continuous, where finitely many pieces suffice and the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1 (again, with finitely many pieces). We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parameterized versions and in the subanalytic setting.
Keywordsp-Adic semi-algebraic functions p-Adic subanalytic functions Lipschitz continuous functions p-Adic cell decomposition
Mathematics Subject Classification (2000)Primary 03C98 12J25 Secondary 03C60 32Bxx 11S80
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