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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 $N=\infty$?

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You don't seem to be using consistent terminology: a hyperplane in linear algebra is a subspace of codimension 1. This doesn't happen if $\,k=\infty\,$ and $\,l,m<\infty\,$ . Equivalently, a hyperplane is a maximal subspace $\,\Longleftrightarrow\,$ the kernel of a non-zero linear functional. Check this carefully and re-edit your question. –  DonAntonio Jan 23 '13 at 15:42
    
@DonAntonio: Sorry about that, I changed the title, I'm interested in hypersurfaces which can have arbitrary codimension. –  okj Jan 23 '13 at 15:45

1 Answer 1

up vote 1 down vote accepted

Use the formula: $$ \dim(V + W) + \dim(V\cap W) = \dim(V) + \dim(W) $$ to find your preferred bounds on dimensions...

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Your first term on the left must be $\,\dim(U+W)\,$...union of subspaces is usually not a subspace. –  DonAntonio Jan 23 '13 at 16:02
    
Thank you! Does this identity have a name or a source so that I can read more about it? –  okj Jan 23 '13 at 20:10
    
proofwiki.org/wiki/… –  Emanuele Paolini Jan 23 '13 at 21:33
    
Thank you! I'm not very familiar with set notation, could you tell me what the difference is between $V+W$ and $V\cup W$? I understand that the latter is the union of the two subspaces, but what is $V+W$? –  okj Jan 23 '13 at 21:56

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