Order stars and stability theorems

This paper clears up to the following three conjectures:

1. The conjecture of Ehle 1] on theA-acceptability of Padé approximations toe ^z , which is true;
2. The conjecture of Nørsett 5] on the zeros of the “E-polynomial”, which is false;
3. The conjecture of Daniel and Moore 2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist.

We further give necessary as well as sufficient conditions forA-stable (acceptable) rational approximations, bounds for the highest order of “restricted” Padé approximations and prove the non-existence ofA-acceptable restricted Padé approximations of order greater than 6. The method of proof, just looking at “order stars” and counting their “fingers”, is very natural and geometric and never uses very complicated formulas.