Re: Questions about Algebraic Functions

John L. Barber wrote:
I have several questions regarding algebraic functions that I'm rather desperate to find the answers to for a paper I'm working on. A search of the literature and the internet doesn't yield anything relevant, so I thought I'd ask here.

I'd previously asked some of these questions in this old thread:

http://mathforum.org/kb/message.jspa?messageID=5659683&tstart=0

My apologies for starting a new thread, but there seemed to be little interest in the old one.

Consider a function f of a (possibly complex) variable z. (For now, let's just consider functions of a single variable.) It is my understanding that f(z) is said to be an algebraic function if there exists a function F(z, w) that is a polynomial in z with coefficients which are polynomials in w, such that F(z, f(z)) = 0. Furthermore, it is my understanding that, given F(z, w), one can find the roots of the original function f(z) by searching for the roots of F(z, 0) = 0.

My first set of questions concerns the nature of the function F(z, w), and my second set concerns the nature of the roots.


* The function F(z, w) *

It is clear that, given an algebraic f(z), the function F(z, w) is not uniquely defined. Multiplying any applicable F(z, w) by any (well, almost any) other function G(z, w) will yield another function H(z, w) = G(z, w)*F(z, w) that also satisfies H(z, f(z)) = 0. Furthermore, there may be roots of H(z, 0) which are not roots of f(z). So my question is:

(1) Does there exist an "optimal" F(z, w) for every algebraic function f(z), such that all of the roots of F(z, 0) are roots of f(z) and vice versa? If so, how do we find and/or identify it?

(2) As an example, consider the function f(z) = sqrt(1+z) + sqrt(1+2z). As I understand it, this is algebraic. However, straightforward algebraic machinations yield F(z, w) = (w^2 - 2 - 3z)^2 - 4(1+z)(1+2z) as an associated F such that F(z, f(z)) = 0. Yet, clearly this F is wrong, since the roots of F(z, 0) = z^2 = 0 are two 0's. 0 is not a root of the original f(z). What have I done wrong, and what is the right F for this f?


* The nature of the roots *

Consider now an algebraic function f such that all of the coefficients contained therein are real. Let's call the vector of parameters (i.e. coefficients) upon which f depends "p". p is a finite list of real numbers. Let's explicitly show the dependence of f upon these parameters: f = f(z, p). For any given value of p, f will have some set of n (possibly complex) roots r(p) = {r1(p), r2(p), ..., rn(p)}. In general, there may be some multiplicity of roots, so that elements of r(p) may be repeated. My questions:

(1) Must n be finite for every p?

(2) Must r(p) be a continuous function of p? In other words, can one of the "root branches" ri(p) jump dicontinuously to another value at some particular value of p?

(3) Must n be the same for every p? In other words, must f(z, p) have the same number of roots, regardless of the values of the coefficients upon which it depends?

It is this last question that concerns me. I have a rather complicated algebraic function which I've been studying the roots of numerically. I've identified 2 "root branches". My problem is that one of these branches seems to stop abruptly at a particular value of the parameter*, so that for a particular regime of parameters there is only one root branch. I'm wondering if this is "allowed" or if I'm just missing a root somehow.

Thanks very much in advance for any help. Even if you don't have time to answer these questions, it would also be helpful to have a pointer to the literature or to a book in which I can find the answers.

*Yes, I remembered to look for complex roots as well.

May be your thread just vanished in all the trash posted at the NG.

I do not quite understand what you really want, but find Lee Rudolph's
answer to be adequate ...

But will try to give a different view, where it does not matter so much
that your function is algebraic - especially since you also want some
dependency on parameters and do not say whether that is algebraic as well,
but let me assume it is analytic (not necessarily a polynom).

Asking for roots of a function g: C -> C is the special case of inveres
values for g. If the function is reasonable one speak of the degree of g by
what is meant the 'generic number of solutions for g(z) = c', c varying.

In case of a polynomial function that is just the degree of the polynom
and if it is exp then it is infinite.

Geometrically it means the number of points in a fibre (if needed then it
is counted with 'multiplicity', just as for z^2=0), call it n.

For a picture it is best to paint it like a 'projection' (from top to bottom).

Now if you have points (of the bottom) where the fibre are exactly n points
then you can choose any and moreover you can extend that to a small neighbour
hood for an isomorphism.

This is one starting point to introduce Riemannian surfaces (surface, since
they are complex, hence 2dim real).

One can show that one can glue this little pieces together and if you do a
Google search for "surface of the square root" you will find some picture
(I once learned this as "a function will find its own surface"). It extends
the function to a larger domain (somewhat abstract, but the picture might
help), it 'lifts' to a function G: X -> C.

Sometimes it is called the natural domain of existence for the function (as
it is - in some sense - the maximal possible domain, where it can live).

Otto Forster's book "Riemannian Surfaces" covers that in some pages (he is
a phantastic teacher, however the book is quite dry).

Now - for example - asking for roots may be seen as to invert G (the extension
of g). If you have your picture at hand you will see that this can not be
done - except you restrict yourself to some parts.

A branch cut exactly does that: take out the stuff sitting cuts the surface
in such pieces, so that you can invert. And that is your 'root'.

It also covers, that a small change gives still a holomorphic solution and
more.

Now for the algebraic case, i.e. your function is a polynomial function and
you even make it depending on (one?) parameter t.

If this deformation through t is 'reasonable' only modifying the coefficients
then: yes, you still are fine - this is called "continouity of roots".

However take care: this does not say, that you easily can do it global, as
continuous means "small changes give well behaviour".

Simple case: x^2 * t + 1 behaves well for t=1 over 0 (2 different solutions)
but everything boils down for t=0 (may become a problem, if you do numerics).

It also may be a problem, if you reach the branch cut (as you vary the surface
through t)

Anyway: you must go through the complex case, Reals do not really help here.


I hope that the above helps a bit and does not contain too many failures by
shorting the actual stuff quite a lot (but I am sure others will point to
errors).


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