Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComputing in quotients of rings of integers
88
0
0.0
(
0
)
Added by
Tommy Hofmann
Publication date
2016
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.
rate research
Read More
Let O be the ring of integers of a number field K. For an O-algebra R which is torsion free as an O-module we define what we mean by a Lambda_O-ring structure on R. We can determine whether a finite etale K-algebra E with Lambda_O-ring structure has an integral model in terms of a Deligne-Ribet monoid of K. This a commutative monoid whose invertible elements form a ray class group.
We present a variation of the modular algorithm for computing the Hermite normal form of an $mathcal O_K$-module presented by Cohen, where $mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996) based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of Cohen to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field $K$.
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve $E$ that runs, under certain heuristics, in time $O((log p)^2p^{1/2})$. The algorithm works by first finding two cycles of a certain form in the supersingular $ell$-isogeny graph $G(p,ell)$, generating an order $Lambda subseteq operatorname{End}(E)$. Then all maximal orders containing $Lambda$ are computed, extending work of Voight. The final step is to determine which of these maximal orders is the endomorphism ring. As part of the cycle finding algorithm, we give a lower bound on the set of all $j$-invariants $j$ that are adjacent to $j^p$ in $G(p,ell)$, answering a question in arXiv:1909.07779.
We show how the minimal free resolution of a set of $n$ points in general position in projective space of dimension $n-2$ explicitly determines structure constants for a ring of rank $n$. This generalises previously known constructions of Levi-Delone-Faddeev and Bhargava in the cases $n=3,4,5$.
One of the many number theoretic topics investigated by the ancient Greeks was perfect numbers, which are positive integers equal to the sum of their proper positive integral divisors. Mathematicians from Euclid to Euler investigated these mysterious numbers. We present results on perfect numbers in the ring of Eisenstein integers.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
