Ideal Bases and Primary Decomposition: Case of Two Variables

A complete structure theorem is given for standard (= Grobner) bases for bivariate polynomials over a field and lexicographical orderings or for univariate polynomials over a Euclidian ring. An easy computation of primary decomposition in such rings is deduced. Another consequence is a natural factorisation of the resultant of two univariate polynomials over the integers which is a generalisation of the ''reduced discriminant'' of a polynomial of degree 2.