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 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 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, 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).
