Canonical simplification on algebraic domains

 

• 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)

• 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

• 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