I am surprised not to have heard mention of magma. In my view this system from Sydney Australia, is the system of choice (ignoring cost) for deeper math. As for the original query: maple or mathematica, I use maple for algebra and GAp for groups (maple is grim here). But in my view maple development should stop. They are getting into thicker mire at each new version. -- I do not buy using ordinary numerical approximation where an exact mathod is called for. This makes some of PARI contentious. It is not mathematics to give probablistic answers without clear warnings! perhaps someone can tell us when numerical approximations are useful? I am used to converting to exact algebraic by using some LLL-type (PSLQ) algorithm - but other cases? John -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.