What can we tell about non-transverse intersection points of (smooth) submanifolds? Especially, in the case of complementary dimensions, how can we define and calculate the ''multiplicity'' of an isolated intersection point, i.e. the number of transverse intersection points collapsed in a nontransverse one? What kind of other invariants of the intersection are known? What can we know about the classification, and what are the obstructions of it?

I guess, some survey literature would be the most helpful for me about this topic.

Here I sketch a reduction of the question which leads to a generalization at the same time.

Without loss off generality we can can assume that

we are in $ \mathbb{R}^n$,

one of the manifolds is a coordinate plane $ K=\mathbb{R}^k$.

the other manifold $L$ is the image of a non-singular map (germ) $ f: (\mathbb{R}^l, 0) \to (\mathbb{R}^n, 0)$.

The intersection point we want to study is the origin.

The first assumtion is satisfied if we study a small neighborhood of the intersection point. The second one can be reached by a choice of the coordinates in $ \mathbb{R}^n$. Note that the second and the third assumptions implicitly contains that the type of the intersection point is meaningful up to ''left-right'' equivalence, i.e. smooth change of the local coordinates on the three submanifolds.

We get a more general question by omitting the nonsingularity assumpion of $f$. Morover, we can also change the first manifold to a map germ $ g: (\mathbb{R}^k, 0) \to (\mathbb{R}^n, 0)$.

Some known related results (assume that $k+l=n$):

If $k=0$, $K$ is the origin, the question reduces to the index of a vector field: How to compute the index of a vectorfield defined by analytic formula?

If $n=2$, $k=1$, $K$ is the $x$-axis, $f$ is non-singular, and the intersection is non-transverse, then locally the image of $f$ is the graph of a germ $h: (\mathbb{R}, 0) \to ( \mathbb{R}, 0) $. Assume that $f$ and $h$ are analytic. Then -- I guess -- the order of the power series of $h$ is the total invariant in some sense.

Globally the intersection number is an invariant: if $K$ and $L$ are closed oriented submanifolds of an oriented manifold $N$, their intersection number is the algebraic number of their transverse intersection points. No matter, how they moved to transverse position, the intersection number depends only on the homology classes represented by $K$ and $L$.

In the complex case the order of the power series of a power series $h: ( \mathbb{C}, 0) \to ( \mathbb{C}, 0) $ is equal to the number of zeros of a stable perturbation of $h$. That is, for $ f: ( \mathbb{C}, 0) \to ( \mathbb{C}^2, 0)$, $f(z)=(z, h(z))$, the number of the transverse intersection points of the $x$-axis in $ \mathbb{C}^2$ and the image of a stable perturbation of $f$ is equal to the order of the power series of $h$.

Questions:

Is there a local version of the intersection number (see 3.), kind of order of tangency? I can see two candidates for it, the first one is the order of the power series, the second one is the algebraic number of intersection points of a stable perturbation. The second one seems meaningful in general -- is it? What does the order of the power series mean in general, which power series should we consider? The two candidates are equal in the complex case (see 4.), but not in the real case (see 2., e.g. $f(x)=(x, x^3)$, the order is $3$, the algebraic number is $1$). What can we tell about these numbers, what is the most general situation?

Let $K= \{x_{k+1}=x_{k+2}=\dots =x_n \} $ and $f=(f_i)_{i=1, \dots, n}$, assumed to be analytic. Then the intersection set is defined by $f(\{f_{k+1}=f_{k+2}=\dots =f_n=0 \})$. It seems that the composition of $f$ with the projection to the $x_{k+1}-x_{k+2}-\dots -x_n$-plane is enough to determine the type of the intersection point, in this way can we reduse the question to the $k=0$ case? Does the ideal generated by $f_i$'s ($i=k+1, \dots, n$), or the factor of the local ring with this ideal have any interesting meaning?

If $k=l$ and $f$ composed with the projection onto the $x_1-\dots-x_k$-plane is a bijection, then the image of $f$ is the graph of a germ $h: (\mathbb{R}^k, 0) \to (\mathbb{R}^k, 0)$. How can we use this germ $h$ to determine the type of the intersection point?

I feel that the question leads naturally to singularity theory, i.e. classifying singularities of germs of maps or spaces, but I cannot see the direct connection: which map or which zero-set should I consider? Another difficulty is that everything is real, not complex.

Order of tangency along directions / curves...