Eigenvalues of $A + \lambda B $ as functions of $\lambda $ Suppose $A $ and $B$ are operators on an $N$-dimensional vector space $V$. For each value of $\lambda$, we have $N $ eigenvalues of $A + \lambda B$. That is, we have a multi-valued function of $\lambda $. 
How about the singularities of this function? Does it have any pole? Generally it will have some branch points, right? In particular, if $A B - BA \neq 0 $, are there necessarily some branch points? 
 A: As has been mentioned in a comment above, you are solving $\det(A+\lambda B-tI)=0$ which is a polynomial of degree $n$ if the matrices are $n \times n$.  In particular, it will be of the form
$$
\sum_{j=0}^n a_j(\lambda)t^j = 0.
$$
If you think about it, $a_n \not=0$.  So, the function that takes $\lambda$ to $t$ is an $n$-valued complex analytic function, and has no singularities.  That is, it really is a function from ${\mathbb C}$ to the $n$-th symmetric power of ${\mathbb C}$.  This means that any symmetric function of all those roots (for example the sum of all the roots) is always an analytic function.  Also, away from the "discriminant set" which are going to be isolated points in the $\lambda$-plane you will locally have $n$ analytic functions $t_1(\lambda), \ldots, t_n(\lambda)$, but you cannot define these functions globally as they get reshuffled as as you go around the points in the discriminant set.
A good reference to look at for the general theory on the geometry of zero sets of such polynomials (or analytic functions), would be some book in algebraic geometry as has already been mentioned.  A reasonable introductory book is Holme's "A Royal Road to Algebraic Geometry".  Also look up "variety" on wikipedia, though wikipedia is not really a good intro.
A book I like very much that is more on the analysis side of things is Hassler Whitney's "Complex Analytic Varieties".  Whitney makes heavy use of the "multivalued function" concept as I mentioned above, which I find very intuitive, so a variety is simply a graph (locally) of a multivalued analytic function.
