# 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)|.

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)|.

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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