  Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T)
the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y].
In a first part, T = p^e is assumed to be the power of an irreducible
polynomial p. A new algorithm that computes a minimal lexicographic Groebner
basis of the ideal ( a, b, p^e), is introduced. A second part extends this
algorithm when T is general through the "local/global" principle realized by a
generalization of "dynamic evaluation", restricted so far to a polynomial T
that is squarefree. The algorithm produces splittings according to the case
distinction "invertible/nilpotent", extending the usual "invertible/zero" in
classic dynamic evaluation. This algorithm belongs to the Euclidean family, the
core being a subresultant sequence of a and b modulo T. In particular no
factorization or Groebner basis computations are necessary. The theoretical
background relies on Lazard's structural theorem for lexicographic Groebner
bases in two variables. An implementation is realized in Magma. Benchmarks show
clearly the benefit, sometimes important, of this approach compared to the
Groebner bases approach.
