Re: Number of roots of an algebraic function

Since you're a poor, ignorant physicist you will be
pardoned if you
don't know that there's a technical meaning for
"algebraic function".

I did know that there was a technical definition, although I wasn't sure if I was using it correctly or not. I got my definition from:

http://mathworld.wolfram.com/AlgebraicFunction.html

Specifically, a (genuine, i.e., not multivalued)

Hmmm, multivalued. My function has terms of the form sqrt(1 + a*z) in it, so I suppose that means it's not single-valued. Unless we stipulate that only a single branch will be "sampled".

function f of a
complex variable z in a domain D in the complex
numbers C is
*algebraic* if there is a complex POLYNOMIAL function
F of two
variables z, w such that F(z,f(z)) = 0 for all z in
D.

I'm not sure if my function satisfies this property or not. It's rather too long to try and write down here. It certainly doesn't have any transcendental functions in it, which is why I have called it "algebraic".

Can I assume that your condition "all of
the parameters
...in the function are real" is equivalent to "there
exists such
an F which is a function with REAL coefficients"?

As I said, I'm not sure about the existence of such an F, although I suspect its existence. I purposely avoided use of the word "coefficient", as it seems to me to imply a multiplicative prefactor that multiplies some power of z.

I'll try and describe my f(z) to make things more clear. My f(z) is a finite sum of terms, where each term is of one of two forms:

1) c*z^m, where c is real, and m is 0, 1, or 2.

2) c*(z^m)*(d*sqrt(1 + a*z) + e*sqrt(1 + b*z)), where a, b, c, d, and e are real, and m is again 0, 1, or 2.

2) What results are there regarding the number of
roots of such
a function f(z)?

We should take the "simplest" F that works for your
given f.
Then write F(z,w) as a polynomial in z with
coefficients that
are polynomials in w; e.g., F(z,w) = (w^2-1)z^3 + wz
+ 1.
A root of the complete algebraic function is a value
z=z_0 such
that F(z_0,0)=0; in this example F(z,0)=-z^3+1, which
has 3
roots, and in general the number of roots (counting
multiplicities,
and making technical corrections when the "leading
coefficient"
is divisible by w) is exactly the highest power of z
that appears
in F(z,w).

Thanks. I've understood this. Is there a formal theorem (with a name) associated with the above statements?

I will search for such an F. The complexity of my f is not so great as to preclude me from finding one by hand.
.

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