Show that the set of values of $w$ is finite

[safwane]

Let $f(x,w,v)$ and $g(x,w,v)$ two polynomial functions (in $x$) with integer coefficients with $x>a$ is an integer variable and $a>0$. Here $w,v$ are also integer variables. Assuming that the degree of $f$ and $g$ are equal and greater than or equal $1$. Let us consider the following **implicit equation**:

[math]w=f(x,w,v)/g(x,w,v)[/math] **holds** for all $x>a$.

Then my **question** is: Show that the set of values of $w$ is finite.

My **solution**:

From this link: https://mathhelpforum.com/threads/eventually-bounded-rational-functions.138510/, it is possible to conclude that the function [math]x→f(x,w,v)/g(x,w,v)[/math] is bounded and hence the set of values of [math]w[/math] is finite.

 

[Dr.Peterson]

You say that f and g are polynomials in x, but you say nothing about how they depend on v and w. And in the equation, you are treating everything, as far as I can see, as functions of w, with x and v as parameters. So I don't see that you can say anything about w.

Even if you were talking about two polynomials in w with the same degree, the rational function would not be bounded; the link refers to its being eventually bounded, because it has a limit: It might have vertical asymptotes, and therefore be unbounded.

What is the actual problem you are working on?

 

[safwane]

So, if no vertical asymptotes, then the set of w is bounded. We can assume that g(x,w,v)>0.

 

[Dr.Peterson]

So, if no vertical asymptotes, then the set of w is bounded. We can assume that g(x,w,v)>0.

On what grounds can you assume that? And is anything known about the dependence on w and v, which I asked about?

The problem still needs a lot of clarification before anyone can answer you.

 

[safwane]

The functions are polynomials in w and v. In the original exercice we can take g(x,w,v)>0.