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We study a monomial derivation $d$ proposed by J. Moulin Ollagnier and A. Nowicki in the polynomial ring of four variables, and prove that $d$ has no Darboux polynomials if and only if $d$ has a trivial field of constants.
Let $k$ be an arbitrary field. In this note, we show that if a sequence of relatively prime positive integers ${bf a}=(a_1,a_2,a_3,a_4)$ defines a Gorenstein non complete intersection monomial curve ${mathcal C}({bf a})$ in ${mathbb A}_k^4$, then the
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $Isubset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting cla
We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Grobner bases, factorization or sub-resultant computations.
Let $R=S/I$ be a graded algebra with $t_i$ and $T_i$ being the minimal and maximal shifts in the minimal $S$ resolution of $R$ at degree $i$. In this paper we prove that $t_nleq t_1+T_{n-1}$, for all $n$ and as a consequence, we show that for Gorenst
Let $K$ be a field and $S = K[x_1,dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible