(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/$C^\infty$ etc.) The classical Implicit Function Theorem starts form the assumption: the matrix of partial derivatives, $F'_y(0,0)$, is non-degenerate.
Often this does not hold. For example, the classical version fails to ensure the solution of equations as simple as $xy=0$ or $y^2=0$ (say over $k[[x]]$).
A strengthening is known as Tougeron's Implicit Function Theorem. Tougeron replaces the condition "$F'_y(0,0)$ is non-degenerate" by the weaker "the entries of $F(x,0)$ belong to $I_{max}(F'_y(x,0))^2$". Here $I_{max}$ is the ideal of the maximal minors, i.e. the zero's fitting ideal.
Tougeron's condition is still too restrictive, e.g. it cannot ensure the solvability of $H(x,y)+y_1 x^m_1+y_2x^m_2+p(x)=0$. (Here $H(x,y)$ is at least quadratic in $y$ while $p(x)$ is of Taylor order at least $2m+1$.)
There have been several strengthenings (i.e. further weakenings of this condition). All of them being sufficient but not necessary.
In arxiv:1311.0088 we have obtained a necessary and sufficient condition for an equation to possess a "good" solution. (Well, actually an order-by-order solution. Once one has it, one gets a solution in the completion. Then one can use some Artin-type approximation results.)
Besides being iff result, this is also quite general, e.g. it holds for finitely generate modules over a commutative associative ring (which is not necessarily over a field). For example, if in the equation $H(x,y)+y_1 x^m_1+y_2x^m_2+p(x)=0$ both $H(x,y)$ and $p(x)$ are defined over $\mathbb{Z}$ (with $p(x)$ of order at least $2m+1$) then we get a solution over $\mathbb{Z}[[x]]$.
- Is anything of this style known? (References?)
- What are the potential applications? (Some particular cases where one needs a solution but $F'_y(0,0)$ is degenerate?)
(For the bookkeeping, here are some related questions: 1, 2, 3, 4)
