Abstract Let $\{{\mathcal{T}}_n\}$ be the bipolar filtration of smooth concordance group topologically slice knots, which was introduced by Cochran et al. It is known that for each $n\ne 1$ ${\mathcal{T}}_n/{\mathcal{T}}_{n+1}$ has infinite rank and ${\mathcal{T}}_1/{\mathcal{T}}_2$ positive rank. In this paper, we show also Moreover, prove there exist infinitely many Alexander polynomials $p(t...

We study primary submodules and decompositions from a differential computational point of view. Our main theoretical contribution is general structure theory representation theorem for an arbitrary finitely generated module over polynomial ring. characterize in terms operators punctual Quot schemes. Moreover, we introduce implement algorithm that computes minimal decomposition module.

In [16], by using an MV-semiring and MV-algebra, we introduced the new definition of MV-semimodule studied some their basic properties. this paper, study present definitions primary ideals MV-semirings, decomposition in A-ideals MV -semimodules, MV-semimodules. Then conditions that A-ideal can have a reduced decomposition.

In Noro (2010) we proposed an algorithm for computing primary ideal decomposition by using the notion of separating ideal and showed that it can efficiently decompose several examples which are hard to decompose by existing algorithms. In particular the number of redundant components produced in the algorithm is zero or very small in many examples, but no theoretical explanation for the efficie...