arXiv subjects
Publications and source records attributed to Joydip Saha.
We study the tangent cone at the origin and the Hilbert series for a family of numerical semigroups generated by concatenation of arithmetic sequences. We prove that all the concatenation classes have Cohen-Macaulay tangent cones except the symmetric class, however, the symmetric class does satisfy Rossi's conjecture.
In this paper, we study the Apery tables for the numerical semigroups given by Bresinsky and Arslan. Using the Apery tables we write the tangent cones of the Bresinsky and Arsalan curves at the origin. Further, we calculate Hilbert series of the tangent cone of the Bresinsky and Arslan curves. We prove that both classes of the curve have Cohen- Macaulay tangent cone.
This paper investigates the projective closure of simplicial affine semigroups in $\mathbb{N}^{d}$, $d \geq 2$. We present a characterization of the Cohen-Macaulay property for the projective closure of these semigroups using Gröbner bases. Additionally, we establish a criterion, based on Gröbner bases, for determining the Buchsbaum property of non-Cohen-Macaulay projective closures of numerical semigroup rings. Lastly, we introduce the concept of $k$-lifting for simplicial affine semigroups in $\mathbb{N}^d$, and investigate its relationship with the original simplicial affine semigroup.
In this paper, we study the class of affine semigroup generated by integral vectors, whose components are in generalised arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on the defining ideal of the semigroup ring for the equality of the Betti numbers of the defining ideal and those of its initial ideal. We introduce the notion of an affine semigroup generated by join of two affine semigroups and show that this affine semigroup exhibits some nice properties including Cohen-Macaulayness.
Our aim in this paper is to compute the Poincaré series of the derivation module of the projective closure of certain affine monomial curves.
We study the projective closures of three important families of affine monomial curves in dimension $4$, namely the Backelin curve, the Bresinsky curve and the Arslan curve, in order to explore possible connections between syzygies and the arithmetic Cohen-Macaulay property.
In this paper, we give the necessary and sufficient conditions for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroups using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also define the nice extension for simplicial affine semigroups, motivated by the notion of a nice extension of the numerical semigroups.
In this paper, we carry out a fairly comprehensive study of special classes of numerical semigroups, and their tangent cones, generated by the sequence of partial sums of an arithmetic progression, in embedding dimension $5$. These classes have unique expansions of the Apery set elements.
In this paper our aim is twofold. First, we introduce the notion of star gluing of numerical semigroups and show that arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure are preserved under this gluing operation. We then give a condition on Gröbner basis of the defining ideal of an affine monomial curve which ensures that the Betti sequence of the affine curve is the same as the Betti sequence of its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct some numerical semigroups, using a gluing technique, such that the Cohen-Macaulay type of corresponding affine curve and its projective closure are both $n$.
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.
Let $K$ be a field and $X$, $Y$ denote matrices such that, the entries of $X$ are either indeterminates over $K$ or $0$ and the entries of $Y$ are indeterminates over $K$ which are different from those appearing in $X$. We consider ideals of the form $I_{1}(XY)$, which is the ideal generated by the $1\times 1$ minors of the matrix $XY$. We prove that the quotient ring $K[X, Y]/I_{1}(XY)$ admits an ASL structure for certain $X$ and $Y$.
In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by the partial sums of a geometric progression, in embedding dimension $4$. Both these classes have the common feature that they have unique expansions of the Apéry set elements.
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.
Let $K$ be a field and $X$, $Y$ denote matrices such that, the entries of $X$ are either indeterminates over $K$ or $0$ and the entries of $Y$ are indeterminates over $K$ which are different from those appearing in $X$. We consider ideals of the form $I_{1}(XY)$, which is the ideal generated by the homogeneous polynomials of degree $2$ given by the $1\times 1$ minors of the matrix $XY$. We prove that $d$-sequences and regular sequences arise naturally as part of generators of $I_{1}(XY)$ for some special cases. We use this information to calculate the equations defining the Rees algebra of $I_{1}(XY)$.
We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviours.
We study numerical semigroups with the property "multiplicity= embedding dimension+1", generated by concatenation of arithmetic sequences.
In this paper we explicitly compute the derivation module of quotients of polynomial rings by ideals formed by the sum or by some other gluing technique. We discuss cases of monomial ideals and binomial ideals separately.
Let $a,n \in \mathbb{Z}^+$, with $a<n$ and $\gcd(a,n)=1$. Let $P_{a,n}$ denote the lattice parallelogram spanned by $(1,0)$ and $(a,n)$, that is, $$P_{a,n} = \left\{ t_1(1,0)+ t_2(a,n) \, : \, 0\leq t_1,t_2 \leq 1 \right\}, $$ and let $$V(a,n) = \# \textrm{ of visible lattice points in the interior of } P_{a,n}.$$ In this paper we prove some elementary (and straightforward) results for $V(a,n)$. The most interesting aspects of the paper are in Section 5 where we discuss some numerics and display some graphs of $V(a,n)/n$. (These graphs resemble an integral sign that has been rotated counter-clockwise by $90^\circ$.) The numerics and graphs suggest the conjecture that for $a\not= 1, n-1$, $V(a,n)/n$ satisfies the inequality $$ 0.5 < V(a,n)/n< 0.75.$$