English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $k$ be an algebraically closed field of characteristic 0, and $A$ aCohen-Macaulay graded domain with $A_0=k$. If $A$ is semi-standard graded(i.e., $A$ is finitely generated as a $k[A_1]$-module), it has the $h$-vector$(h_0, h_1, ..., h_s)$, which encodes the Hilbert function of $A$. From now on,assume that $s=2$. It is known that if $A$ is standard graded (i.e.,$A=k[A_1]$), then $A$ is level. We will show that, in the semi-standard case,if $A$ is not level, then $h_1+1$ divides $h_2$. Conversely, for any positiveintegers $h$ and $n$, there is a non-level $A$ with the $h$-vector $(1, h,(h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings(equivalently, normal toric rings).

Non-level semi-standard graded Cohen-Macaulay domain with $h$-vector $(h_0,h_1,h_2)$