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Powers of (monomial) ideals is a subject that still calls attraction in various ways. In this paper we present a nice presentation of high powers of ideals in a certain class in K [ x 1 , ... , x n ] and K [ [ x 1 , ... , x n ] ]. As an interesting application it leads to an algorithm for computation of the Ratliff–Rush operation on ideals in that class. The Ratliff–Rush operation itself has several applications, for instance, if I is a regular m -primary ideal in a local ring (R , m) , then the Ratliff–Rush associated ideal I ˜ is the unique largest ideal containing I with the same Hilbert polynomial as I. [ABSTRACT FROM AUTHOR]

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R / I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R / I. This is exploited in this paper to introduce the use of two special (non-monomial) types of basis functions with nice properties. This allows us, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments. [ABSTRACT FROM AUTHOR]