Papers by Reza Abdolmaleki
International Journal of Algebra and Computation, May 28, 2024
Let A be a commutative Noetherian local ring with maximal ideal m, and let I be an ideal. The fib... more Let A be a commutative Noetherian local ring with maximal ideal m, and let I be an ideal. The fiber cone is then an image of the polynomial ring over the residue field A/m. The kernel of this map is called the defining ideal, and it is natural to ask how to compute it. In this paper, we provide a construction for the defining ideals of Cohen-Macaulay fiber cones.
Archiv der Mathematik, Mar 24, 2021
This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial r... more This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound µ(I 2 ) ≥ 9 for the number of minimal generators of I 2 with µ(I) ≥ 6. Recently, Gasanova constructed monomial ideals such that µ(I) > µ(I n ) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that µ(I) > µ(I 2 ) > • • • > µ(I n ) = (n + 1) 2 for any positive integer n, which provides one of the most unexpected behaviors of the function µ(I k ). The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of R/I n descends.
Archiv der Mathematik
This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial r... more This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound µ(I 2 ) ≥ 9 for the number of minimal generators of I 2 . Recently, Gasanova constructed monomial ideals such that µ(I) > µ(I n ) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that µ(I) > µ(I 2 ) > • • • > µ(I n ) = (n + 1) 2 for any positive integer n, which provides one of the most unexpected behaviors of the function µ(I k ).
Communications in Algebra
Kyoto Journal of Mathematics, Sep 1, 2022
Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, ... more Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese type ideals I. Using this formula, we show that sat(I k) is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
arXiv (Cornell University), Dec 24, 2022
Let S = K[x1,. .. , xn] the polynomial ring over a field K. In this paper for some families of mo... more Let S = K[x1,. .. , xn] the polynomial ring over a field K. In this paper for some families of monomial ideals I ⊂ S we study the minimal number of generators of I k. We use this results to find some other Betti numbers of these families of ideals for special choices of n, the number of variables.
arXiv (Cornell University), Jun 7, 2021
Given any equigenerated monomial ideal I with the property that the defining ideal J of the fiber... more Given any equigenerated monomial ideal I with the property that the defining ideal J of the fiber cone F (I) of I is generated by quadratic binomials, we introduce a matrix such that the set of its binomial 2-minors is a generating set of J. In this way, we characterize the fiber cone of sortable and Freiman ideals.
Kyoto Journal of Mathematics
Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, ... more Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese type ideals I. Using this formula, we show that sat(I k) is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
arXiv (Cornell University), Sep 24, 2019
Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, ... more Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese type ideals I. Using this formula, we show that sat(I k) is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
arXiv: Commutative Algebra, 2020
Given a number $q$, we construct a monomial ideal $I$ with the property that the function which d... more Given a number $q$, we construct a monomial ideal $I$ with the property that the function which describes the number of generators of $I^k$ has at least $q$ local maxima.
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Papers by Reza Abdolmaleki