Re: Algebraic Generating Functions - Closure Properties Question

In article <47c3bc01_1@xxxxxxxxxxxxxxxxxxxxxx>,
"Eric Fennessey" <eric.fennessey@xxxxxxxxxxxxxx> wrote:

Folk,

I am looking at the set of functions Y in Z[[X]] that satisify some equation
of the form,

F_0 + F_1*Y + F_2*Y^2 + ... + F_n*Y^n = 0,

For some F_i's in Z[X] not all 0.

Can someone point me to a proof (if it's true) that this set is closed under
addition and multiplication? In particular, given Y and Z algebraic, I
would appreciate any suggestions on how to construct a polynomial that has
Y+Z as a root and one that has Y*Z as a root, given polynomials for Y and Z.

I don't know if it works, but here's what I'd try first.

Consider the simpler situation where you're looking
for complex numbers Y that satisfy a polynomial with
integer coefficients - in other words, looking for
algebraic numbers. Given two algebraic numbers
Y, Z (better: given the polynomials they satisfy),
there is a simple construction for a polynomial
satisfied by Y + Z, similarly for YZ. It involves
the resultant.

What works in the simpler context may work for you, too.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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