Let a=(a_0, a_1, ..., a_n), b=(b_0, b_1, ..., b_n) that belong to R^{n+1}. Define polynomials f_a(t)=a_0+a_1t+ ... + a_nt^n and f_b(t)=b_0+b_1t+ ... + b_nt^n and let f_{ab}(t)=f_a(t)f_b'(t)-f_a'(t)f_b(t)=c_0+c_1t+ ... + c_{2n-2}t^{2n-2}, where ' denotes derivative. From the above setting we may define the (bilinear) map F:R^{2n+2} \to R^{2n-1}, F(a,b)=c.

QUESTION: Is F onto?

If so, is it known what the fiber F^{-1}(c) will look like?