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8
votes
2answers
4k views

Fast trace of inverse of a square matrix

Which would be the most efficient way (in computational time) to compute tr(inv(H)), where H is a (dense) square matrix? In my particular problem I also have a LU decomposition of H already ...
12
votes
6answers
1k views

Seeking Noetherian normal domain with vanishing Picard group but not a UFD

Once again, the question says it all. My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
6
votes
3answers
2k views

Reduction from factoring to solving Pell equation

The paper Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem claims There are reductions from factoring to solving Pell’s equation, and from solving Pell’s ...
7
votes
3answers
796 views

$\omega(p^n - 1)$ as $n \rightarrow \infty$

Although I am also interested in the number of distinct prime factors (not counting multiplicity), today I use $\omega(m)$ to denote the number of (positive) prime factors (with multiplicity) of the ...
6
votes
2answers
255 views

Cubic graphs whose 2-factors all have the same cycle type

Let $G$ be a bridgeless cubic graph. I am interested in such graphs where all 2-factors are isomorphic (as graphs), i.e. have the same partition as cycle type. We'll say that this partition is ...
5
votes
1answer
202 views

Large gaps between consecutive irreducible polynomials with small heights

For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then ...
0
votes
0answers
65 views

Is this factoring algorithm sufficiently efficient for some integers of special kind?

Basically the question is if $m$ is factored over the integers, can it be relatively efficiently factored over $\mathbb{Z}[\sqrt{n}]$ where $n$ is not factored, but might be of special form? Suppose ...