# Genus Zero Diophantine Equations and Infinite Valuations

I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:

[1] Solving genus zero diophantine equations with at most two infinite valuations by Dimitrios Poulakis and Evaggelos Voskos

[2] Bounds for the size of integral points on curves of genus zero by Dimitrios Poulakis

It turns out that $f(X, Y)=0$ might actually have infinitely many integer solutions if the set of infinite valuations, $C_\infty$, satisfies certain properties. The set $C_\infty$ is defined as follows:

Definition. Let $f(X,Y)$ be an absolutely irreducible polynomial with integer coefficients such that the curve defined by the equation $f(X,Y)=0$ is of genus 0. We denote by $C$ the projective curve defined by $F(X,Y,Z)=0$, where $F(X,Y,Z)$ is the homogenization of $f(X,Y)$. Let $Q$ be an algebraic closure of the field of rational numbers $Q$ and $Q(C)$ be the function field of $C$. If $P$ is a point on $C$, we denote by $O_P(C)$ the local ring at $P$. We call, as usual, the points $(x:y:0)$ on $C$ points at infinity and we denote by $C_\infty$ the set of discrete valuation rings $U$ of $Q(C)$ such that $U$ dominates the local ring $O_P(C)$ at a point $P$ at infinity (i.e. $U$ contains $O_P(C)$ and the maximal ideal of U contains the maximal ideal of $O_P(C)$).

My question is, this definition of $C_\infty$ seems very abstract to me and I don't how to compute it for my curve. It seems that in some cases it is equal to the number of points at infinity. I can't find any book that talks about this $C_\infty$ either. I would be grateful if anyone can provide more information on this $C_\infty$, or even better provide a way to compute it for any genus zero curve.

Edit: This definition comes from the above papers by Poulakis. The same definition appeared again in the paper "Affine curves with infinitely many integral points" (MR1949864) by Poulakis. In this paper, he provided counterexamples to a previous result by Joseph Silverman (MR1765971) and mentioned that the result can be corrected if we replace "points at infinity" with "discrete valuation rings at infinity". Poulakis also stated that $C_\infty$ is essentially the desingularization of the points at infinity.

• Where does this definition of $C_{\infty}$ come from? Is it from the Poulakis papers? Can you give bibliographic details on those papers? Commented Mar 16, 2018 at 0:09
• I think we're talking about MR1890582 Poulakis, Dimitrios; Voskos, Evaggelos; Solving genus zero Diophantine equations with at most two infinite valuations, J. Symbolic Comput. 33 (2002), no. 4, 479–491 and MR1925357 Poulakis, D. Bounds for the size of integral points on curves of genus zero. Acta Math. Hungar. 93 (2001), no. 4, 327–346. Commented Mar 16, 2018 at 0:13