Homogeneous Buchberger algorithms and Sullivant's computational commutative algebra challenge

学习 Buchberger 算法的性能度量

Learning a performance metric of Buchberger's algorithm

本文利用易计算的理想生成元统计量构建线性回归与递归神经网络，仅凭初始输入预测 Buchberger 算法的多项式加法次数，证明该性能度量可被机器学习。

Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

Let $K\ < x_i\ >$ be the free associative algebra generated by a finite or countable number of variables $x_i$. The notion of "letterplace correspondence" introduced in [1,2] for the graded (two-sided) ideals of $K\ < x_i\ >$ is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid $\mathbb N$ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of $K\ < x_i,t\ >$, wh…

Macaulay bases of modules

We define Macaulay bases of modules, which are a common generalization of Groebner bases and Macaulay $H$-bases to suitably graded modules over a commutative graded $\mathbf{k}$-algebra, where the index sets of the two gradings may differ. This includes Groebner bases of modules as a special case, in contrast to previous work on Macaulay bases of modules. We show that the standard results on Groebner bases and Macaulay $H$-bases generalize in fields of arbitrary characteristic to Macaulay bases, including the reduction algorithm and Buchberger's criterion and algorithm. A key result is that Ma…