# Tagged Questions

**2**

votes

**0**answers

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

**2**answers

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

**3**answers

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 + ...