arXiv subjects

Sumita Garai

Publications and source records attributed to Sumita Garai.

5 recordsLinked to original sources

arXiv2026

Rational linear subspaces of hypersurfaces over finite fields

Let $X \subset \mathbb{P}^n$ be a hypersurface of degree $d$ defined over a finite field of characteristic $p > 0$. We prove that if $n \ge r + \binom{d+r}{r+1}$, then $X$ contains a rational $r$-plane. We prove better bounds when $X$ is smooth and $p$ is sufficiently large. We also present experimental data regarding the existence of rational lines on cubic threefolds over $\mathbb{F}_7$, $\mathbb{F}_8$, and $\mathbb{F}_9$. In particular, we construct an example of a smooth cubic threefold over $\mathbb{F}_7$ with exactly $8$ rational lines. It remains an open question whether smooth cubic threefolds over $\mathbb{F}_7$, $\mathbb{F}_8$, and $\mathbb{F}_9$ always contain a rational line.

math.AG↗

arXiv2026

Project Auto-World: Towards Automated Benchmarking of Neural Relational Reasoners

Reasoning about relational structures remains a significant challenge for neural models, particularly when they must systematically apply learned knowledge to problem instances that are harder than those seen in training. Progress is hampered by the difficulty of evaluating such generalization, since a priori, it is rarely clear what makes an instance hard. We study how this issue can be addressed by using large language models (LLMs) to automate benchmark generation, learning to produce increasingly challenging instances in an end-to-end manner. Concretely, given a world parametrized by Datalog rules, and an Edge Transformer as the reasoning evaluator, we use LLM-driven evolutionary search (based on FunSearch) and autonomous agentic search to discover sampling functions that yield hard problem instances. We also show that the Edge Transformer can be improved using this data such that it generalizes well to further data perturbations. Finally, we show that the same machinery can be applied to novel worlds proposed by LLMs, opening the door to autonomous research on neural relational reasoning.

cs.AI↗

arXiv2024

A principled framework to assess the information-theoretic fitness of brain functional sub-circuits

In systems and network neuroscience, many common practices in brain connectomic analysis are often not properly scrutinized. One such practice is mapping a predetermined set of sub-circuits, like functional networks (FNs), onto subjects' functional connectomes (FCs) without adequately assessing the information-theoretic appropriateness of the partition. Another practice that goes unchallenged is thresholding weighted FCs to remove spurious connections without justifying the chosen threshold. This paper leverages recent theoretical advances in Stochastic Block Models (SBMs) to formally define and quantify the information-theoretic fitness (e.g., prominence) of a predetermined set of FNs when mapped to individual FCs under different fMRI task conditions. Our framework allows for evaluating any combination of FC granularity, FN partition, and thresholding strategy, thereby optimizing these choices to preserve important topological features of the human brain connectomes. By applying to the Human Connectome Project with Schaefer parcellations at multiple levels of granularity, the framework showed that the common thresholding value of 0.25 was indeed information-theoretically valid for group-average FCs despite its previous lack of justification. Our results pave the way for the proper use of FNs and thresholding methods and provide insights for future research in individualized parcellations.

q-bio.NC↗

arXiv2019

Computing endomorphism rings and Frobenius matrices of Drinfeld modules

Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.

math.NT↗

arXiv2019

Endomorphism rings of reductions of Drinfeld modules

Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $ϕ$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $ϕ$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $ϕ\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$. The endomorphism ring $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(ϕ\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$. Let $\mathcal{O}_\mathfrak{p}$ be the integral closure of $A$ in $K$, and let $π_\mathfrak{p}\in \mathcal{E}_\mathfrak{p}$ be the Frobenius endomorphism of $ϕ\otimes\mathbb{F}_\mathfrak{p}$. Then we have the inclusion of orders $A[π_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$ in $K$. We prove that if $\mathrm{End}_{F^\mathrm{alg}}(ϕ)=A$, then for arbitrary non-zero ideals $\mathfrak{n}, \mathfrak{m}$ of $A$ there are infinitely many $\mathfrak{p}$ such that $\mathfrak{n}$ divides the index $χ(\mathcal{E}_\mathfrak{p}/A[π_\mathfrak{p}])$ and $\mathfrak{m}$ divides the index $χ(\mathcal{O}_\mathfrak{p}/\mathcal{E}_\mathfrak{p})$. We show that the index $χ(\mathcal{E}_\mathfrak{p}/A[π_\mathfrak{p}])$ is related to a reciprocity law for the extensions of $F$ arising from the division points of $ϕ$. In the rank $r=2$ case we describe an algorithm for computing the orders $A[π_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$, and give some computational data.

math.NT↗