algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Computational number theory; algorithms; computer algebra systems; power series; solution of linear differential equations; monodromy groups; continued fraction expansion; computations with abelian varieties; SCRATCHPAD II; elliptic curve computations; primality testing; factorization; periods of abelian integrals; algebraic relations; transcendence; Galois groups; bibliography 