Curve with one place at infinity
Definition
Let \(F = y^n + a_1(x)y^{n-1} + \cdots + a_n(x)\) be a nonzero polynomial of \(\mathbb{K}[x][y]\) and assume, possibly after a change of variables, that \(deg_x ~ a_i(x) < i\) for all \(i \in \{1, 2, \ldots, n\}\) such that \(a_i(x) \ne 0\). Let \(C = V(F)\) be the plane algebraic curve in \(\mathbb{A}_{\mathbb{K}}^2\) defined by \(F(x,y) = 0\), and let \(h_F(u,x,y) = u^nF(\frac{x}{u}, \frac{y}{u})\). The projective curve \(V(h_F)\) is the projective curve closure of \(C\) in \(\mathbb{P}_{\mathbb{K}}^2\). The embedding \(\mathbb{A}_{\mathbb{K}}^2 \subset \mathbb{P}_{\mathbb{K}}^2\) that send \((x,y) \to (u : x : y)\) has been used. By hypothesis on the coefficients of \(F\), \((0 : 1 : 0)\) is the unique point of \(V(h_F)\) at the line at infinity \(u = 0\). It is said that \(F\) has one place at infinity if \(h_F\) is analytically irreducible at \((0 : 1 : 0)\).