At the end of my last post I posed a problem to which I don’t know the solution, and which am very curious to see how our readers might tackle. It occurs to me that, if I want people to think about a problem, I shouldn’t put it at the end of a long post, below the fold. So
Problem Let and be polynomials in two variables such that and exist. Show that the determinant
is not a nonzero constant. (Equivalently, show that it has a zero for some .)
One warning: the zero is not always on the hyperbola . For example, if then the determinant is
which does not vanish on the hyperbola.