Approximations and Lipschitz continuity in padic semialgebraic and subanalytic geometry
Abstract
It was already known that a padic, locally Lipschitz continuous semialgebraic function is piecewise Lipschitz continuous, where the pieces can be taken semialgebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1. We do this by proving the following fine preparation results for padic semialgebraic 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 parametrized versions and in the subanalytic setting.
 Publication:

arXiv eprints
 Pub Date:
 September 2010
 arXiv:
 arXiv:1009.3414
 Bibcode:
 2010arXiv1009.3414C
 Keywords:

 Mathematics  Logic;
 Mathematics  Algebraic Geometry;
 03C98;
 32Bxx;
 12J25 (Primary);
 03C60;
 11S80 (Secondary)
 EPrint:
 12 pages