Powers of (monomial) ideals is a subject that still calls attraction in various ways.be a monomial ideal and let G(I) denote the (unique) minimal monomial generating set of I. How small can |G(I i )| be in terms of |G(I)|? We expect that the inequality |G(I 2 )| > |G(I)| should hold and that |G(I i )|, i ≥ 2, grows further whenever |G(I)| ≥ 2. In this paper we will disprove this expectation and show that for any n and d there is an m-primary monomial ideal I ⊂ K[x 1 , . . . , x n ] such that |G(I)| > |G(I i )| for all i ≤ d.

We propose a novel approach to distinguish table vs non-table ideals by using different machine learning algorithms. We introduce the reader to table ideals, assuming some knowledge on commutative algebra and describe their main properties. We create a data set containing table and non-table ideals, and we use a feedforward neural network model, a decision tree and a graph neural networks for the classification. Our results indicate that there exists an algorithm to distinguish table ideals from notable ideals, and we prove it along some novel results on table ideals.

Let R be a Cohen–Macaulay local K-algebra or a standard graded K-algebra over a field K with a canonical module $\omega _R$ . The trace of $\omega _R$ is the ideal $\operatorname {tr}(\omega _R)$ of R which is the sum of those ideals $\varphi (\omega _R)$ with ${\varphi \in \operatorname {Hom}_R(\omega _R,R)}$ . The smallest number s for which there exist $\varphi _1, \ldots , \varphi _s \in \operatorname {Hom}_R(\omega _R,R)$ with $\operatorname {tr}(\omega _R)=\varphi _1(\omega _R) + \cdots + \varphi _s(\omega _R)$ is called the Teter number of R. We say that R is of Teter type if $s = 1$ . It is shown that R is not of Teter type if R is generically Gorenstein. In the present paper, we focus especially on zero-dimensional graded and monomial K-algebras and present various classes of such algebras which are of Teter type.

No abstract

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let I & K½x 1 , :::, x n be a monomial ideal and let G(I) denote the (unique) minimal monomial generating set of I. How small can jGðI i Þj be in terms of jGðIÞj? Until recently, it was widely expected that jGðI 2 Þj ! jGðIÞj would always hold. The first counterexamples emerged in 2018 for n ¼ 2. In this article we show that for any n and d there is an m-primary monomial ideal I & K½x 1 , :::, x n such that jGðIÞj > jGðI i Þj for all i d: ARTICLE HISTORY