arXiv subjects

Manav Batavia

Publications and source records attributed to Manav Batavia.

6 recordsLinked to original sources

arXiv2026

The arithmetic rank of nullcones of classical invariant rings

We compute the arithmetic rank of nullcone ideals arising from natural actions of the special linear, orthogonal, and symplectic groups on direct sums of copies of their standard and dual representations. Over an infinite field of characteristic different from two, we show that the arithmetic rank equals the Krull dimension of the invariant ring, and compute this dimension. In positive characteristic, the required lower bounds are not detected by local cohomology, and are obtained using \'etale cohomology.

math.AC↗

arXiv2026

Elimination Templates in Macaulay2

We introduce the package \texttt{EliminationTemplates} for the Macaulay2 computer algebra system, which provides tools for constructing automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent parameters. This article provides a self-contained description of how elimination templates are constructed for such families and their specialization properties. Additionally, we describe the main functionality and datatypes provided by our package, and illustrate its usage on several examples, including applications from computer vision from which elimination templates originated.

math.AC↗

arXiv2026

Vanishing of local cohomology in unramified mixed characteristic

Given an ideal $I$ in a regular local ring $A$, the cohomological dimension of $I$ in $A$ is the index of the highest non-vanishing local cohomology of $A$ supported at $I$. Determining effective upper bounds on the cohomological dimension in terms of topological invariants of $\text{Spec}(A/I)$ is a central problem in commutative algebra. In equal characteristic, Faltings proved in 1980 a general bound on the cohomological dimension of an ideal in terms of its big height. In this article, we extend Faltings' result to the unramified mixed characteristic setting and show that the resulting bound is sharp.

math.AC↗

arXiv2025

The arithmetic rank of the residual intersections of a complete intersection ideal

The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual intersection of an ideal generated by $n$ indeterminates for all $m\geq n$ and in every characteristic. We further give an explicit description of its set-theoretic generators. Our main result provides a sharp upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in any Noetherian local ring. In particular, given a complete intersection ideal of height at least two, any of its generic residual intersections -- including its generic link -- fails to be a set-theoretic complete intersection in characteristic zero.

math.AC↗

arXiv2023

Syzygies of associated graded modules

Given a finitely generated module $M$ over a Noetherian local ring $R$, we give a characterization for the first syzygy of the associated graded module $G_{\mathfrak{m}}(M)$ to be equigenerated. As an application of this, we identify a complex of free $G_{\mathfrak{m}}(R)$-modules, arising from given free resolution of $M$ over $R$, which is a resolution of $G_{\mathfrak{m}}(M)$ if and only if $G_{\mathfrak{m}}(M)$ is a pure $G_{\mathfrak{m}}(R)$-module. We also give several applications of the purity of $G_{\mathfrak{m}}(M)$. Our results demonstrate that while not all algebraic properties of a module carry over to its associated graded module, the purity of the minimal free resolution of $G_{\mathfrak{m}}(M)$ ensures that several important invariants are inherited. In addition, we provide sufficient conditions for Cohen-Macaulayness and purity of $G_{\mathfrak{m}}(M)$, and provide a local version of the Herzog-K\"uhl equations.

math.AC↗

arXiv2019

Fermat's polygonal number theorem for repeated generalized polygonal numbers

In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized $m$-gonal numbers required to represent every positive integer and we furthermore generalize this result to obtain optimal bounds when many of the generalized $m$-gonal numbers are repeated $r$ times, where $r\in\mathbb{N}$ is fixed.

math.NT↗