On a regularity-conjecture of generalized binomial edge ideals
Article, Collectanea Mathematica, 2025, DOI Link
View abstract ⏷
In this paper, we prove the upper bound conjecture proposed by Saeedi Madani and Kiani on the Castelnuovo–Mumford regularity of generalized binomial edge ideals. We give a combinatorial upper bound of regularity for generalized binomial edge ideals, which is better than the bound claimed in that conjecture. Also, we show that the bound is tight by providing an infinite class of graphs.
On the Depth of Generalized Binomial Edge Ideals
Article, Mediterranean Journal of Mathematics, 2024, DOI Link
View abstract ⏷
This research focuses on analyzing the depth of generalized binomial edge ideals. We extend the notion of d-compatible map and use it to give a combinatorial lower bound for the depth of generalized binomial edge ideals. Subsequently, we determine an upper bound for the depth of generalized binomial edge ideals in terms of the vertex-connectivity of graphs. We demonstrate that the difference between the upper and lower bounds can be arbitrarily large, even in cases when one of the bounds is sharp. In addition, we calculate the depth of generalized binomial edge ideals of certain classes of graphs, including cycles and graphs with Cohen-Macaulay binomial edge ideals.
Unboundedness of the first Betti number and the last Betti number of numerical semigroups generated by concatenation
Mehta R., Saha J., Sengupta I.
Article, Indian Journal of Pure and Applied Mathematics, 2024, DOI Link
View abstract ⏷
We show that the minimal number of generators and the Cohen-Macaulay type of a family of numerical semigroups generated by concatenation of arithmetic sequences is unbounded.
Moh’s example of algebroid space curves
Mehta R., Saha J., Sengupta I.
Article, Journal of Symbolic Computation, 2021, DOI Link
View abstract ⏷
In this paper we revisit the family of algebroid space curves defined by Moh and find an explicit minimal generating set for the defining ideal.
Numerical semigroups generated by concatenation of arithmetic sequences
Mehta R., Saha J., Sengupta I.
Article, Journal of Algebra and its Applications, 2021, DOI Link
View abstract ⏷
We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviors.
Betti numbers of Bresinsky’s curves in 4
Mehta R., Saha J., Sengupta I.
Article, Journal of Algebra and its Applications, 2019, DOI Link
View abstract ⏷
Bresinsky defined a class of monomial curves in 4 with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behavior of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for the defining ideal of this class of curves.
Unboundedness of Betti numbers of curves
Mehta R., Saha J., Sengupta I.
Article, ACM Communications in Computer Algebra, 2018, DOI Link
View abstract ⏷
Bresinsky defined a class of monomial curves in A 4 with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behaviour of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for this class. We also propose a general construction of such curves in arbitrary embedding dimension.
