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In this paper, we attempt to resolve the singularities of the zero variety ofa $C^{\infty}$ function of two variables as much as possible by using ordinaryblowings up. As a result, we formulate an algorithm to locally express the zerovariety in the ``almost'' normal crossings form, which is close to the normalcrossings form but may include flat functions. As an application, weinvestigate analytic continuation of local zeta functions associated with$C^{\infty}$ functions of two variables. As is well known, thedesingularization theorem of Hironaka implies that the local zeta functionsassociated with real analytic functions admit the meromorphic continuation tothe whole complex plane. On the other hand, it is recently observed that thelocal zeta function associated with a specific (non-real analytic) $C^{\infty}$function has a singularity different from the pole. From this observation, thefollowing questions are naturally raised in the $C^{\infty}$ case: how wide themeromorphically extendible region can be and what kinds of informationessentially determine this region? This paper shows that this region can bedescribed in terms of some kind of multiplicity of the zero variety of each$C^{\infty}$ function. By using our blowings up algorithm, it suffices toinvestigate local zeta functions in the almost normal crossings case. This casecan be effectively analyzed by using real analysis methods; in particular, avan der Corput-type lemma plays a crucial role in the determination of theabove region.

Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions