The following question is an attempt to find a lower bound for the value of a polynomial at integer points. It is something that I originally thought about while trying to understand how it would be possible to approach [this MO question](https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers) about polynomials representing the nonnegative integers, but does seem to be very interesting in its own right. I guess this would fall under the study of Diophantine approximation/geometry, which I am not at all expert on. So it could even be a known conjecture or theorem, or obviously false. Maybe someone on MO will be able to say?


As is well known, the [Thue-Siegel-Roth theorem](http://en.wikipedia.org/wiki/Thue%E2%80%93Siegel%E2%80%93Roth_theorem) says that, for an irrational algebraic number $\alpha$, there are only finitely many pairs of integers $p,q$ satisfying the inequality $\vert p/q-\alpha\vert\le q^{-2-\epsilon}$. Here, $\epsilon$ is any fixed positive real number.
This is easily seen to be equivalent to the following lower bound on the growth of homogeneous and irreducible polynomials $f\in\mathbb{Q}[X,Y]$ of degree $d > 1$. For all but at most finitely many integer pairs $x,y$, the inequality 
$$
\vert f(x,y)\vert\ge\vert y\vert^{d-2-\epsilon}\qquad\qquad{\rm(1)}
$$
holds. The argument is very simple. We can decompose $f(x,y)$ as $y^d\prod\_{i=1}^d(x/y-\alpha\_i)$ for distinct irrational algebraic numbers $\alpha\_i$. As $x/y-\alpha\_i$ can only be made arbitrarily small for at most one $i$ at a time, (1) is equivalent to applying the Thue-Siegel-Roth theorem to $x/y-\alpha\_i$.


Now for my question. Is there an extension of (1) to non-homogeneous polynomials $f$? Now, I realize that it cannot possibly carry directly across to the non-homogeneous case in the same form. For one thing, a simple change of variables allows us to replace $f$ by a polynomial of arbitrarily large degree, such as $\tilde f(x,y)=f(x+y^r,y)$, which would invalidate any inequality depending on the degree of $f$ and, similarly, the right hand side of (1) would change form under changes of variables. To guess how this can be fixed, we can look at [Siegel's theorem](http://en.wikipedia.org/wiki/Siegel%2527s_theorem_on_integral_points), which says that $f(x,y)=a$ has only finitely many integer solutions in $x,y$ whenever $f-a$ defines a curve over $\mathbb{Q}$ of genus $g$ at least one. It seems reasonable then, that a generalization of (1) should involve the genus $g$ of the curve defined by $f-a$, for typical rationals $a$, and not the degree.


So, to be precise, my question is whether there is an increasing and unbounded function $\phi\colon\mathbb{N}\to\mathbb{R}$ with the following property: If $f\in\mathbb{Q}[X,Y]$ is such that $f-a$ defines a curve of genus $g$ (for all but finitely many $a$), there exists a nonconstant polynomial $h\in\mathbb{Q}[X,Y]$ such that the inequality
$$
\vert f(x,y)\vert\ge \vert h(x,y)\vert^{\phi(g)-\epsilon}\qquad\qquad{\rm(2)}
$$
holds on $\mathbb{Z}\times\mathbb{Z}$ outside of a finite set (\*). We might even hope that $h(x,y)=y$ under a change of variables, but that seems like a bit much to ask. Comparing with the case where $f$ is homogeneous of degree d, the genus of $f-a$ for nonzero $a$ is given by $g=(d-1)(d-2)/2$ and we have $\phi(g)=(-1+\sqrt{1+8g})/2$, although I doubt that precise form would hold in the non-homogeneous case.


(\*) Edit: It is easy to construct polynomials which vanish or degenerate into something simpler on a given finite set of curves. E.g., $f=\prod\_{i=1}^n(X-a\_i)$ is zero on the curves $x=a\_i$. So I should say that (2) holds on $\mathbb{Z}\times\mathbb{Z}$ outside of a finite set of curves (of genus zero).



Bounding the growth of rational bivariate polynomials from below
What you are asking is for a strong effective form of Siegel's theorem. There are a number of conjectures in this direction, all considered very hard. For example, if $f=x^3-y^2$, what you are asking is essentially Hall's conjecture: <http://en.wikipedia.org/wiki/Hall%27s_conjecture> .
In general, it follows from Vojta's conjectures (LNM 1239). Nothing close is known. For certain classes of polynomials (general cubics, $y^m - f(x)$, a few others) you can get a lower bound of the type $\log \log \log \max\{x,y\}$ give or take a log or two, from the theory of linear forms in logarithms.


