Lexicographic Gröbner bases of bivariate polynomials modulo a univariate one

Let T(x)∈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)∈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 Gröbner 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 Gröbner basis computations are necessary. The theoretical background relies on Lazard's structural theorem for lexicographic Gröbner bases in two variables. An implementation is realized in Magma. Benchmarks show clearly the benefit, sometimes important, of this approach compared to the Gröbner bases approach.