Gin and Lex of certain monomial ideals

Abstract

Let $A=K[x_1,...,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ of characteristic 0 with each $°x_i=1$. Given arbitrary integers $i$ and $j$ with $2\le i\le n$ and $3\le j\le n$, we will construct a monomial ideal $I\subset A$ such that (i) $β_k(I) < β_k(\Gin(I))$ for all $k<i$, (ii) $β_i(I) = β_i(\Gin(I))$, (iii) $β_\ell(\Gin(I)) < β_\ell(\Lex(I))$ for all $\ell <j$ and (iv) $β_j(\Gin(I)) = β_j(\Lex(I))$, where $\Gin(I)$ is the generic initial ideal of $I$ with respect to the reverse lexicographic order induced by $x_1>>...>x_n$ and where $\Lex(I)$ is the lexsegment ideal with the same Hilbert function as $I$.

9 pages, minor grammatical changes

References in corpus (1)

• Reduction numbers and initial ideals