arXiv subjects
Publications and source records attributed to Gaurab Tripathi.
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 we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.
In this paper we prove conditions for transversal intersection of monomial ideals and derive a simplicial characterization of this phenomenon.
In this paper we compute Gröbner bases for determinantal ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are both matrices whose entries are indeterminates over a field $K$. We use the Gröbner basis structure to determine Betti numbers for such ideals.
In this paper we propose a model for computing a minimal free resolution for ideals of the form $I_{1}(X_{n}Y_{n})$, where $X_{n}$ is an $n\times n$ skew-symmetric matrix with indeterminate entries $x_{ij}$ and $Y_{n}$ is a generic column matrix with indeterminate entries $y_{j}$. We verify that the model works for $n=3$ and $n=4$ and pose some statements as conjectures. Answering the conjectures in affirmative would enable us to compute a minimal free resolution for general $n$.
In this paper we derive some conditions for transversal intersection of polynomial ideals. We exhibit some examples. Finally, as an application of the results proved, we compute the Betti numbers for ideals of the form $I_{1}(XY) + J$, where $X$ and $Y$ are matrices and $J$ is the ideal generated by the $2\times 2$ minors of the matrix consisting of any two rows of $X$.
Let $\mm=(m_0,m_1,m_2,n)$ be an almost arithmetic sequence, i.e., a sequence of positive integers with ${\rm gcd}(m_0,m_1,m_2,n) = 1$, such that $m_0<m_1<m_2$ form an arithmetic progression, $n$ is arbitrary and they minimally generate the numerical semigroup $Γ= m_0\N + m_1\N + m_2\N + n\N$. Let $k$ be a field. The homogeneous coordinate ring $k[Γ]$ of the affine monomial curve parametrically defined by $X_0=t^{m_0},X_{1}=t^{m_1},X_2=t^{m_3},Y=t^{n}$ is a graded $R$-module, where $R$ is the polynomial ring $k[X_0,X_1,X_3, Y]$ with the grading $°{X_i}:=m_i, °{Y}:=n$. In this paper, we construct a minimal graded free resolution for $k[Γ]$.