If two monic polynomials have no common roots, are the coefficients of their product locally diffeomorphic to the product of the coefficients?

Question

Let $P^d (t,\lambda)$ be the "generic" d-th degree monic polynomial $P^d (t,\lambda) = t^d + \sum\limits_{i=1}^d \lambda_i t^{d-i}$ with real coefficients. Let $\lambda(\xi,\eta)$ be given by the equation $P^d(t,\lambda(\xi,\eta)) = P^l (t,\xi) P^{d-l}(t,\eta)$. If $P^l(t,\xi)$ and $P^{d-l}(t,\eta)$ have no common roots (in $\mathbb C$, i.e. $P^l(t,\xi)$ and $P^{d-l}(t,\eta)$ have no common factors), is $\lambda$ a local diffeomorphism at $(\xi,\eta)$? Is there an easy way to see this?

Answer 1

Ah, it turns out it was worth learning some basics about resultants. The matrix $d_{(\eta,\xi)} \lambda$ is exactly the Sylvester matrix for the polynomials $P^l(t,\xi)$ and $P^{d-l}(t,\eta)$. This matrix represents a system of linear equations for finding polynomials $A$ and $B$ so that $A(t)P^l(t,\xi) + B(t)P^{d-l}(t,\eta) = 0$, $deg(A) < d-l$ and $deg(B) < l$. The polynomials $P^l(t,\xi)$ and $P^{d-l}(t,\eta)$ have a common factor if and only if there exist such $A$ and $B$ not both equal to zero. This occurs exactly when the determinant of the Sylvester matrix is zero. If the determinant of the Sylvester matrix is non-zero, then $P^{l}(t,\xi)$ and $P^{d-l}(t,\eta)$ have no common factor. But if the determinant of $d_{(\xi,\eta)} \lambda$ is non-zero, then the map $\lambda$ is locally a diffeomorphism near $(\xi,\eta)$ by the inverse function theorem.