An extension of Sturm's theorem to two dimensions (Q1357061)

Let \(f(x,y)\) be a square-free polynomial and let \(C\) be the set of points at which \(f(x,y)= 0\). Let \(D\) be a closed rectangle with sides parallel to the coordinate axes and let \(BD\) denote the boundary of \(D\). \(D'\), the interior of \(D\), is \(D\backslash BD\). To find if \(C\cap BD=\emptyset\) one can use Sturm's theorem. If \(C\cap BD=\emptyset\) then the author shows that if \(C\cap D'\neq\emptyset\) there exists a point \(x_0\), \(y_0\) in \(D'\) at which \(f\) and \({\partial f\over\partial x}\) are both zero. Elimination of \(x\) gives a polynomial in \(y\) which is zero at \(y_0\), and Sturm's Theorem enables one to decide if such a point exists.