16 research outputs found
An Algorithm for Computing the Ratliff-Rush Closure
Let I\subset K[x,y] be a -primary monomial ideal where K is a field.
This paper produces an algorithm for computing the Ratliff-Rush closure I for
the ideal I= whenever m_{i} is contained in the integral closure
of the ideal . This generalizes of the work of Crispin
\cite{Cri}. Also, it provides generalizations and answers for some questions
given in \cite{HJLS}, and enables us to construct infinite families of
Ratliff-Rush ideals
Reduced Gr\"obner Bases of Certain Toric Varieties; A New Short Proof
Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of
positive integers. Let C be a toric variety in the affine (n+1)-space, defined
parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a
minimal Gr\"obner basis for the toric ideal which is the defining ideal of C
and give sufficient and necessary conditions for this basis to be the reduced
Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a
much simpler proof than that of \cite{Ayy}
Normality of Monomial Ideals
Given the monomial ideal I=(x_1^{{\alpha}_1},...,x_{n}^{{\alpha}_{n}})\subset
K[x_1,...,x_{n}] where {\alpha}_{i} are positive integers and K a field and let
J be the integral closure of I . It is a challenging problem to translate the
question of the normality of J into a question about the exponent set
{\Gamma}(J) and the Newton polyhedron NP(J). A relaxed version of this problem
is to give necessary or sufficient conditions on {\alpha}_1,...,{\alpha}_{n}
for the normality of J. We show that if {\alpha}_{i}\epsilon{s,l} with s and l
arbitrary positive integers, then J is normal
Monomial ideals with tiny squares and Freiman ideals
summary:We provide a construction of monomial ideals in such that , where denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on that generalize some results of\/ S. Eliahou, J. Herzog, M. Mohammadi Saem (2018), J. Herzog, M. Mohammadi Saem, N. Zamani (2019), and J. Herzog, A. Asloob Qureshi, M. Mohammadi Saem (2019)