Computing characteristic sets of bounded unions of polynomial ideals

The surprising fact that Hilbert's basis theorem in algebra shows identifiabilty of ideals of polynomials in the limit from positive data is derived by the correspondence between ideals and languages in the context of machine learning. This correspondence also reveals the difference between the two and raises new problems to be solved in both of algebra and machine learning. In this article we solve the problem of providing a concrete form of the characteristic set of a union of two polynomial ideals. Our previous work showed that the finite basis of every polynomial ideal is its characteristic set, which ensures that the class of ideals of polynomials is identifiable from positive data. Union or set-theoretic sum is a basic set operation, and it could be conjectured that there is some effective method which produces a characteristic set of a union of two polynomial ideals if both of the basis of ideals are given. Unfortunately, we cannot find a previous work which gives a general method for how to find characteristic sets of unions of languages even though the languages are in a class identifiable from positive data. We give methods for computing a characteristic set of the union of two polynomial ideals.