List of $s, t$ where $|f(s, t)|<M$

by didgogns   Last Updated August 01, 2020 11:20 AM - source

I am given a large homogeneous polynomial of rational coefficients $f(s, t)$ and very large $M$. I want to find list of $(s, t)$ which satisfies $|f(s, t)|<M$.

My try:

Let $t=0$. There is only one nonzero term now and the problem is easy.

Now let $x=\frac{s}{t}$. Now we can represent $f(s, t)$ as $f(x)$ and the problem is finding list of rational $x$ satisfying $|f(x)|<\frac{M}{t^{d}}$, where $d$ is degree of $f$.

Assume $t=1$. Now $x$ is integer and it is possible to find a list of $x$ satisfying $|f(x)|<M$.

Assume $t=2$. Now it is possible to find a list of half-integer $x$ satisfying $|f(x)|<\frac{M}{2^{d}}$.

I could go on this way, but I'm not sure whether there exists a bound $B$ where it can be proven that I don't need to consider $t>B$ and if there is, how can I estimate it.

Any better approach or idea to estimate B?

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Updated March 08, 2019 15:20 PM