Rational Functions 3
The critical points of a rational function r(t) = f(t)/g(t) are the zeroes of its derivative
r '(t)   =   f ' g - g' f
    g2
That is, critical points are the zeroes of f ' g - g' f, the Wronskian of f and g.

    Rational functions f(t)/g(t) and h(t)/k(t) are equivalent if the linear span of f and g equals the linear span of h and k.

Theorem (Eremenko-Gabrielov, 2002)
A rational function with only real critical points is equivalent to a real rational function.