arXiv subjects

Isaac Rajagopal

Publications and source records attributed to Isaac Rajagopal.

6 recordsLinked to original sources

arXiv2026

Universality for cokernels of partially random integral matrices

Given any $\varepsilon > 0$, let $M(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$, with all entries independent and $\varepsilon$-balanced (lying in each residue class mod $p$ with probability at most $1-\varepsilon$). Wood proved that as $n \to \infty$ the distribution of $\mathrm{cok}(M(n))$ approaches Cohen and Lenstra's conjectured distribution of class groups. Given $\alpha,\beta >0$ such that $\alpha + \beta <1$, we prove that the distribution of $\mathrm{cok}(M(n))$ still approaches the Cohen--Lenstra distribution even if we weaken the hypothesis by allowing up to $\alpha n$ entries per column and up to $\beta n$ entries per row of $M(n)$ to not be $\varepsilon$-balanced. We also weaken the independence condition by allowing certain types of dependence between the entries of each column. In addition, we prove that, for any $\delta > 0$, the cokernels of random band matrices of width $\log(n)^{1+\delta}$ with $\varepsilon$-balanced entries in the band and arbitrary entries outside of it will also approach the Cohen--Lenstra distribution, which answers a question of Kang--Lee--Yu.

math.PR↗

arXiv2026

The discrete logarithm problem in cokernels of $\mathcal{O}_K$-matrices

In 2009 and 2010, Blackburn and Shokrieh independently found that the discrete logarithm can be computed efficiently on the sandpile group of a graph, meaning that sandpile groups are not secure for cryptography. We generalize this problem to cokernels of matrices with entries in the ring of integers $\mathcal{O}_K$ of a number field $K$. When $K$ has nontrivial class group, the failure of the Euclidean algorithm in $\mathcal{O}_K$ is an obstacle to generalizing previous methods. For $M$ in $\mathrm{M}_{n\times m}(\mathcal{O}_K)$, we overcome this obstacle to efficiently compute discrete logarithms in $\mathrm{cok}(M) = \mathcal{O}_K^n/M\mathcal{O}_K^m$. In particular, we find an algorithm with time complexity $\tilde{O}((m+n)^{\omega+1})$, where $\omega$ is an exponent of matrix multiplication, to compute discrete logarithms in $\mathrm{cok}(M)$ when $\mathrm{cok}(M)$ is viewed either as an $\mathcal{O}_K$-module or as a group. When $M$ is Hermitian with respect to a Galois involution $\sigma$ and nonsingular, we improve the time complexity to $\tilde{O}(n^\omega)$.

math.NT↗

arXiv2025

Uniform bounds on periodic points of polynomials with good reduction

We establish effective bounds on the number of periodic points of degree-$d$ polynomials $\phi$ defined over $p$-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials $X^d + c$ with $c$ integral at some prime dividing $d$. As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields $K$, giving the explicit uniform bound $\#\mathrm{Per}_K(\phi) \leq d^{[K:\mathbb{Q}]}$.

math.NT↗

arXiv2025

Possible Sizes of Sumsets

Nathanson introduced the range of cardinalities of $h$-fold sumsets $R(h,k) := \{|hA|:A \subset \mathbb{Z} \text{ and }|A| = k\}.$ Following a remark of Erd\H{o}s and Szemer\'edi that determined the form of $R(h,k)$ when $h=2$, Nathanson asked what the form of $R(h,k)$ is for arbitrary $h, k \in \mathbb{N}$. For $h \in \mathbb{N}$, we prove there is some constant $k_h \in \mathbb{N}$ such that if $k > k_h$, then $R(h,k)$ is the entire interval $\left[hk-h+1,\binom{h+k-1}{h}\right]$ except for a specified set of $\binom{h-1}{2}$ numbers. Moreover, we show that one can take $k_3 = 2$.

math.CO↗

arXiv2024

Variations on five-dimensional sphere packings

We analyze Sz\"oll\H{o}si's recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We construct five-dimensional sphere packings from these configurations, which augment Conway and Sloane's list of conjecturally optimal packings. We also construct a new kissing configuration in nine dimensions. None of these constructions improves on the known records, but they provide geometrically distinct constructions achieving these records.

math.MG↗

arXiv2024

The Proportion of Irreducible p-adic Polynomials

We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational functions in $p$. This relates to previous work done to find exact proportions of $p$-adic polynomials of degree $n$ which have $k$ roots.

math.NT↗