Bivariate systems of polynomial equations with roots of high multiplicity.

Given a bivariate system of polynomial equations with fixed support sets A , B it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity i for all i in the range { 0 , 1 , ... , | A | − | conv (A) ⊖ B | − 1 } , where A ⊖ B is the set of all integral vectors that shift B to a subset of A. As an application, we classify all pairs (A , B) such that the system supported at (A , B) does not have a solution of multiplicity higher than 2. [ABSTRACT FROM AUTHOR]