Finite Sets of Affine Points with Unique Associated Monomial Order Quotient Bases

The quotient bases for zero-dimensional ideals are often of interest in the investigation of multivariate polynomial interpolation, algebraic coding theory, and computational molecular biology, etc. In this paper, we discuss the properties of zero-dimensional ideals with unique monomial quotient bases, and verify that the vanishing ideals of Cartesian sets have unique monomial quotient bases. Furthermore, we reveal the relation between Cartesian sets and the point sets with unique associated monomial quotient bases.