Given polynomial $f(x,y)$ with integer coefficients, may be reducible, but without linear factors. For positive integer $n$ denote by $a_n$ the number of points $(x,y)\in \frac1n \mathbb{Z}^2$ on a curve $f(x,y)=0$. May it appear that $a_n$ tends to infinity when $n$ increases?, but is always finite? Similar question: if we consider only bounded part of our curve, say $\{(x,y):f(x,y)=0,x^2+y^2<R^2\}$, and define $b_n$ as the cardinality of the intersection of this set with lattice $\frac1n \mathbb{Z}^2$, may $b_n$ tend to infinity?

Given polynomial $f(x,y)$ with integer coefficients, may be reducible, but without linear factors. For positive integer $n$ denote by $a_n$ the number of points $(x,y)\in \frac1n \mathbb{Z}^2$ on a curve $f(x,y)=0$. May it appear that $a_n$ tends to infinity (when $n$ increases taking all positive integer values), but is always finite? Similar question: if we consider only bounded part of our curve, say $\{ (x,y):f(x,y)=0,x^2+y^2 < R^2 \}$, and define $b_n$ as the cardinality of the intersection of this set with lattice $\frac1n \mathbb{Z}^2$, may $b_n$ tend to infinity?

Given polynomial $f(x,y)$ with integer coefficients, may be reducible, but without linear factors. For positive integer $n$ denote by $a_n$ the number of points $(x,y)\in \frac1n \mathbb{Z}^2$ on a curve $f(x,y)=0$. May it appear that $a_n$ tends to infinity when $n$ increases?

Given polynomial $f(x,y)$ with integer coefficients, may be reducible, but without linear factors. For positive integer $n$ denote by $a_n$ the number of points $(x,y)\in \frac1n \mathbb{Z}^2$ on a curve $f(x,y)=0$. May it appear that $a_n$ tends to infinity when $n$ increases?, but is always finite? Similar question: if we consider only bounded part of our curve, say $\{(x,y):f(x,y)=0,x^2+y^2<R^2\}$, and define $b_n$ as the cardinality of the intersection of this set with lattice $\frac1n \mathbb{Z}^2$, may $b_n$ tend to infinity?