In modeling physical systems it is sometimes useful to construct border bases of 0-dimensional polynomial ideals which are contained in the ideal generated by a given set of polynomials. We define and construct such subideal border bases, provide som
e basic properties and generalize a suitable variant of the Buchberger-Moeller algorithm as well as the AVI-algorithm to the subideal setting. The subideal version of the AVI-algorithm is then applied to an actual industrial problem.
