- for polynomials, it is necessary to define a term ordering--often
the ordering of polynomial variables is sufficient and terms are
ordered according to the variables present and their degrees
- for rational functions
- canonical simplification of numerator and denominator
- dividing numerator and denominator by their greatest common
divisor
- ensuring that the leading coefficient (coefficient of the first
term of a polynomial in a given ordering) satisfies some
condition (e.g., making sure that the leading coefficient of the
denominator of a rational function is positive)

- canonical simplification of numerator and denominator
- for rational functions extended by rational exponents (radicals)
- unnested radicals (a radical cannot be inside another radical):
there exist theories for simplifying these forms
- nested radicals: the theories are very complicated--it is
possible to make use of the same methods as for algebraic functions

- unnested radicals (a radical cannot be inside another radical):
there exist theories for simplifying these forms
- algebraic functions: simplifications are performed "modulo" the
Groebner basis of the system of polynomials defining the algebraic
functions
- elementary transcendental functions and transcendental functions:
there exist structural theorems--these are very complicated

Richard Liska