Branch points of an equation

Question

I was reading the Mathematica Guidebook to Symbolics by Michael Trott, and saw the following expression to evaluate the branch points of a Riemann surface given by $f(x,a)=x^3+x^2+ax-1/2$.

Eliminate[{#==0},D[#,x]==0},x]&[eq=x^3+x^2+ax-1/2]

This gives the result $36a-4a^2+16a^3+19=0$. The roots of this equation are the branch points of $f(x,a)$.

My question is: Why is this so? How does eliminating $x$ from $f(x,a)=0$ and $\dfrac{df(x,a)}{dx}=0$ lead us to the solution?

NOTE: I'm not asking for an explanation of the Mathematica command... I don't get the math behind it.

Answer 1

Imagine the plane $(x,a)$ where the equation $f(x,a)=0$ gives a curve $C$. We project the curve onto the $a$-axis. Branch points are the points on the $a$ axis over which the projection "is not so nice". You can imagine e.g. the points over which the projection "folds", e.g. if $f(x,a)=x^2-a$ then $a=0$ is such a point.

Now you have to imagine, though, that $a$ and $x$ are complex numbers (and hence $a$ is a point in a plane, and $C$ is 2-dimensional). Over a general point $a$ the equation $f(x,a)=0$ has $n$ different solutions (for $x$), where $n$ is the degree of $f$ as a polynomial in $x$ (in your case $n=3$). We have $n$ different solutions iff $f(x,a)=0$ and $df(x,a)/dx=0$ (for this given $a$) don't have a common root. Over such points $C$ covers the plane of $a$'s in $n$ sheets. When, however, $f(x,a)=0$ and $df(x,a)/dx=0$ do have a common root, the number of roots of $f(x,a)=0$ is less than $n$, so over these points $a$ some of the sheets meet. These are the branch points. In fact, what happens, if you go around such a point then you would see that you get from one sheet (also called one "branch") to another. That's why these points are called branching.

(I skipped some unimportant technical details)

Answer 2

Branch points will occur at values of $x$ which are double or higher roots of $f(x,a)=0$, that is, where $f(x,a)$ and $f_x(x,a)$ have common zeros. "Elimination of x from $f(x,a)$ and $f_x(x,a)$" amounts to running the Euclidean algorithm on $f$ and $f'$, where the field of coefficients is rational functions of $a$ with complex coefficients. The expression the algorithm produces is the (non-zero, constant) GCD, unless $f(x,a)$ and $f_x(x,a)$ have a common factor for every $a$, as in $f(x,a)=(x-a)^2$. Avoiding that kind of degeneracy, for only finitely-many values of $a$ that GCD vanishes, which shows a non-trivial GCD.