As we know, Newton polygons could be used to determine the Puiseux series of algebraic curves (see, for example, Kirwan's Complex Algebraic Curves, chapter 7). Different branches correspond to different Puiseux series, then the asymptotic behavior of such a branch could be seen from the Puiseux series. However, I need to determine the asymptotic behavior of a general curve $F(x,y)=0$ near $(0,0)$ where $F\colon\mathbb R^2\to\mathbb R$ is a smooth ($\mathcal C^\infty$) function and $F(0,0)=0$. Suppose that the Taylor series of $F$ is $\sum_{j,k\ge0}a_{jk}x^jy^k\in\mathbb R[[x,y]]$ (might be divergent), and the Newton polygon is depicted as the lower convex hull of the degrees of nonzero terms: $\{\,(m,n)\,\colon\,a_{mn}\neq0\,\}$. Can we still claim that the asymptotic behavior ($y\sim x^\alpha$, etc) of branches of the curve $F(x,y)=0$ near the singular point $(0,0)$ follows from such a Newton diagram?

As a very simple case, if $\partial F/\partial x,\partial F/\partial y$ don't vanish at $(0,0)$, then the asymptotic behavior of $F(x,y)=0$ near $(0,0)$ is (non-degenerately) linear, following from the implicit function theorem, which coincides the information given by the Newton diagram.


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