2
votes
0answers
103 views

A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad) Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
6
votes
2answers
612 views

Implicit function theorem at a singular point?

Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ ...
3
votes
3answers
478 views

Closure of singular points

Let $f(x,y)$ be a complex degree $d$ polynomial that has this particular form. $$ f = \frac{f_{02}}{2} y^2 + \frac{f_{21}}{2} x^2 y + \frac{f_{12}}{2} x y^2 + \frac{f_{03}}{6} y^3 + ...