Complete Residue Systems: A Primer and an Application

Pietro Paparella

Home/Archive/arXiv

arXiv · 1206.0486

Complete Residue Systems: A Primer and an Application

Pietro Paparella

View original source ↗Read source document ↗Export metadata ↓

Abstract

Complete residue systems play an integral role in abstract algebra and number theory, and a description is typically found in any number theory textbook. This note provides a concise overview of complete residue systems, including a robust definition, several well-known results, a proof to the converse of a well-known theorem, ancillary results pertaining to an application arising from the study of the roots of nonnegative matrices, and extends our knowledge of complete residue systems in relation to complete sets of roots of unity.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Pietro Paparella. 2012-08-17. Complete Residue Systems: A Primer and an Application. https://arxiv.org/abs/1206.0486

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The degrees of irreducible factors of binomials and multiplicativity of the n-Hartley condition

We prove that the degrees of irreducible factors of a binomial $t^n-α$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the $mn$-Hartley condition if and only if it has the $m$-Hartley and $n$-Hartley conditions.

math.NT↗

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗

arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗