Cusp-type singularities of real analytic curves in the complex plane

where a, b are real-valued real analytic functions defined near the origin. We always assume that 0 ∈ C, and that all changes of coordinates are local and fix the origin. We say that two parameterized curves C and C̃ are equivalent if there exist a biholomorphic map h and a real analytic diffeomorphism φ so that h(C(t)) = C̃(φ(t)). We also say that C, C̃ are formally equivalent, if h is a formal biholomorphic map and φ is a formal real power series with φ(0) 6= 0. Without parametrization, we shall see that (1.1) defines an irreducible real analytic set in C, or an analytic arc having the origin as the end point. An obvious invariant is the smallest integer n so that C(t) = γ(t) for some real analytic γ(t) with γ(0) 6= 0. Here the classification for γ is restricted to biholomorphic maps commuting with νn : z → ez. We shall see, in next section, that classifications for the parameterized curves C that are not arcs, for the unparametrized curves C, and for the liftings γ are determined by each other, when the curves are not arcs. Some simple modifications will suffice to reduce the classification of arcs to that of non arcs. We first formulate some results for smooth real analytic curves in C, under a change of holomorphic coordinates commuting with ν = νn. Let γ be a smooth real analytic curve, and τ = τγ be the anti-holomorphic reflection of γ. Associated with ν the indicator of γ is χ = ντντ . We start with