Mixed matrices and binomial ideals Original Research Article

Given an integral vector u /gE Zn, one may associate with it the binomial /tfu = Xu+ − Xu− in Z[X] = Z[X1, …, Xn] where u+ and u− are the positive and negative supports of u, respectively. We say that u is mixed if u+,u− ≠ 0 and a matrix M is mixed if all its rows are mixed. We investigate relationships between the matrix M whose rows are u1, …, ur and the ideal I = left angle bracket/tfu1, …, /tfu1right-pointing angle bracket. For example, if M contains no square mixed submatrix of any size, then the vectors u1, …, ur are linearly independent and /tfu1, …, /tfur form a regular sequence in Z[X]. This allows us to decide if a semigroup ring is a complete intersection. When applied to numerical semigroups, the results give an alternate proof of a theorem by Delorme which characterizes numerical semigroups that are complete intersections.