A Theory of Linear Fractional Transformations of Rational Functions

If g, g are complex rational functions, we say that g ∼ g if g = ( ax+b cx+d )−1 ◦ g ◦ (ax+b cx+d ) , where ∣∣∣∣ a b c d ∣∣∣∣ 6= 0. For practical purposes, the general problem of finding a collection of rational invariants that are sufficient to partition ∼ into equivalency classes may be intractable for arbitrary degree rational functions. In this paper, we first outline a simple and naive meta-method for finding weak rational invariants when g and g satisfy g ∼ g. These ‘weak’ invariants can be combined to create ‘strong’ invariants. This meta-method makes the invariants seem almost self-evident. We now know that there is a very large number of these weak rational invariants which we divide into two levels. We apply this meta-method by finding three first level invariants that hold for arbitrary degree rational functions. Then we give alternate proofs using the well-known theory of resultants. These proofs are on the same level as the theorems themselves. In some special cases such as when ax+b cx+d = ax + b, a linear function, our methods yield a large number of first level invariants which can also be extended to an infinite number. Also, by giving the reader one single axiom, our methods can be completely understood by a naive person. This is in sharp contract to the classical theory of invariants (see [2]) which is very specialized. At the end, we state necessary and sufficient conditions so that Ax 2+Bx+C Hx2+DX+E ∼ Ax2+Bx+C Hx2+Dx+E . The applications in this paper deal exclusively with first level invariants. However, we have computed and independently verified nine second level invariants for rational quadratics. This gives a total of 12 weak invariants for rational quadratics, and each of these invariants has a different meaning. These 12 invariants cannot possibly be independent. As an obvious extension of our methods, we also state an unproven algorithm which computes from scratch