We study the formerly established concept of deformation of a planar curve and clarify its applicability and range. We present several applications on classical curves.

We consider typically real harmonic univalent functions in the unit disk 𝔻 whose range is the complex plane slit along infinite intervals on each of the lines x ± ib, b > 0. They are obtained via the shear construction of conformal mappings of 𝔻 onto the plane without two or four half-lines symmetric with respect to the real axis.

We are interested on families of formal power series in $\text{(}{ℂ}^{},0\text{)}$ parameterized by ${ℂ}^n$ ($\widehat{f}={\sum }_{m=0}^{\infty }{P}_m(x_1,\cdots ,x_n)x^m$). If every $P_m$ is a polynomial whose degree is bounded by a linear function ($deg{P}_m\le Am+B$ for some $A\>0$ and $B\ge 0$) then the family is either convergent or the series $\widehat{f}(c_1,\cdots ,c_n,x)\notin ℂ\left\{x\right\}$ for all $(c_1,\cdots ,c_n)\in {ℂ}^n$ except a pluri-polar set. Generalizations of these results are provided for formal objects associated to germs of diffeomorphism (formal power series, formal meromorphic functions, etc.). We are interested on describing the nature of the set of parameters where...