On Computing the Elimination Ideal Using Resultants with Applications to Gr\"obner Bases

Resultants and Gr\"obner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the bivariate case, in which the elimination ideal is principal. We study - by means of elementary tools - the difference between the multiplicity of the factors of the generator of the elimination ideal and the multiplicity of the factors of the resultant.Comment: 7 page

A polyhedral approach to computing border bases

Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is \emph{independent} from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with \emph{preference} on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree. The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces