Let V be a vector space of dimension $\text{dim}\left(V\right)=k$, where $k$ can be $\infty$. If I have two hypersurfaces in V of dimensions $\text{dim}\left(S_1\right)=l$ and $\text{dim}\left(S_2\right)=m$, respectively. What is the dimension of their intersection, i.e. $\text{dim}\left(S_1\cap S_2\right)=?$
I am interested in the case where $k$ is finite and also the case where $k$ is infinite.
Furthermore, say I have $N$ hypersurfaces with respective dimensions $d_1,d_2,…,d_N$. What is the dimension of their intersection?
It seems that most likely the solutions to these questions will be of the form, $\text{dim}\left(S_1\cap S_2\right)\leq k-\left(l+m\right)$, i.e. that there isn't a single answer but rather an upper bound on the dimensionality of the intersection, if so, that is what I'm after.
To summarize:
1) What is $\text{dim}\left(S_1\cap S_2\right)$ when $k$ is finite?
2) What is $\text{dim}\left(S_1\cap S_2\right)$ when $k$ is infinite?
3) What is the answer to (1) when there are $N<\infty$ hypersurfaces intersecting?
4) What is the answer to (2) when there are $N<\infty$ hypersurfaces intersecting?
5) What are the answers to (3-4) when $N=\infty$?
Best Answer
Use the formula: $$ \dim(V + W) + \dim(V\cap W) = \dim(V) + \dim(W) $$ to find your preferred bounds on dimensions...