Canonical Bases for Subalgebras on two Generators in the Univariate Polynomial Ring

In this paper we examine subalgebras on two generators in the univariate polynomial ring. A set, S, of polynomials in a subalgebra of a polynomial ring is called a canonical basis (also referred to as SAGBI basis) for the subalgebra if all lead monomials in the subalgebra are products of lead monomials of polynomials in S. In this paper we prove that a pair of polynomials {f, g} is a canonical basis for the subalgebra they generate if and only if both f and g can be written as compositions of polynomials with the same inner polynomial h for some h of degree equal to the greatest common divisor of the degrees of f and g. Especially polynomials of relatively prime degrees constitute a canonical basis. Another special case occurs when the degree of g is a multiple of the degree of f . In this case {f, g} is a canonical basis if and only if g is a polynomial in f . 1 Canonical bases for subalgebras When studying subalgebras of the polynomial ring it is important to construct convenient bases which can be used for example to determine whether a given element is in the subalgebra. Given a finite set of generators for an ideal it is algorithmic to construct a so-called Gröbner basis for the ideal which has this property. The concept of SAGBI basis, where SAGBI is an abbreviation for Subalgebra Analog to Gröbner Bases for Ideals, was introduced by Kapur and Madlener ([?]) and independently by Robbiano and Sweedler ([?]). They also present a method for constructing such bases given a set of generators for a subalgebra of a multivariate polynomial ring. In general this method