Resolution of \(P(x,y)=0\) (Q1182361)

Let \(P\) be a polynomial in two variables with coefficients in \(\mathbb{R}\). The author considers the problem of tracing the set \(C=\{(x,y)\in\mathbb{R}\mid P(x,y)=0\hbox{ and }| x|\leq 1\}\). \(C\) splits into \(m+n\) connected components; the first \(m\) have endpoints on the boundary of the pair of lines \(| x|=1\), and the last \(n\) are closed curves inside the band \(| x| \leq 1\). An approach, based on the solution of a system of ordinary differential equations, is presented. This approach provides all the components of the set \(C\) at once.