On the ring $R=F[x_1,\cdots ,x_n]$ of polynomials in n variables over a field $F$ special isomorphisms $A$’s of $R$ into $R$ are defined which preserve the greatest common divisor of two polynomials. The ring $R$ is extended to the ring $S=F{\left[[x_1,\cdots ,x_n]\right]}^{+}$ and the ring $T=F\left[[x_1,\cdots ,x_n]\right]$ of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms $A$’s are extended to automorphisms $B$’s of the ring $S$. Using the property that the isomorphisms $A$’s preserve GCD it...

We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal $I$ is a CM Stanley ideal, then $I^p$ is a Stanley ideal as well, where $I^p$ is the polarization of $I$.

In this note we study some algebraic properties of Borel Ideals in the ring of polynomials over an effective field of characteristic zero by using a suitable partial order relation defined on the set of terms of each degree. In particular, in the three variable case, we characterize all the 0-dimensional Borel ideals corresponding to an admissible $h$-vector and their minimal free resolutions.

Résumé. Soit f un polynôme à deux indéterminées. On appelle nombre de Łojasiewicz à l'infini de f le nombre de Łojasiewicz à l'infini de son application gradient. Dans cet article nous montrons tout d'abord que l'on peut calculer le nombre de Łojasiewicz d'un polynôme à partir des diagrammes de Eisenbud et Neumann de toutes les courbes f(x,y) = t. Ensuite nous montrons que l'on peut définir un nombre de Łojasiewicz intrinsèque en prenant le maximum des nombres de Łojasiewicz de f ∘ ϕ si f est bon...

Let $\Delta$ be a pure simplicial complex on the vertex set $\left[n\right]=\{1,...,n\}$ and $I_{\Delta }$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,...,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$ and this is equivalent to saying that $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{\left(m\right)}$ is not (pretty) clean for all integer $m\ge 3$. If $dim\left(\Delta \right)=1$, we also prove that $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$) is clean if and only if $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$,...