Questions tagged [euclidean-domain]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
47 views

some superharmonic function as a universal lower bound on Lipschitz domains

Question: for any bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$, does there always exist a nonnegative function $\phi\in C^2(\Omega)$ such that $\phi$ vanishes on $\partial\Omega$ the normal ...
1 vote
1 answer
83 views

Can a polynomial be evaluated from evaluations of partial interpolations? (Or: can the unique solution of congruences be written in a certain way?)

Formally, let $K$ be a field, $a \in K[x]$, $D \subseteq K$, and $E = \{(x_i, a(x_i)) \mid x_i \in D\}$ be distinct evaluations of $a$ where $\lvert E\rvert > \operatorname{deg}(a)$ (so $E$ ...
  • 539
9 votes
1 answer
307 views

Is $\mathbb{Z}[i,\varphi]$ a Euclidean domain?

I've already asked this question on Math StackExchange but having gotten no responses this may be more obscure than I had initially believed. Here $\varphi=\frac{1+\sqrt{5}}{2}$. It's true that $\...
  • 417
2 votes
0 answers
43 views

Rings with terminating division chains of a given length

Let $R$ be an integral domain. Given $a,b\in R$, then a division chain for $(a,b)$ is a sequence where we take $r_{-1}=a$, $r_0=b$, and for each $n>0$ we take $r_n=r_{n-1}s_n+r_{n-2}$ for some $...
  • 15.5k
3 votes
0 answers
134 views

For which $n$ is this ring an euclidean domain?

Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$. Question: Is $A_n$ for all $n$ an euclidean domain? Is there a good choice for an euclidean function? ...
  • 22.7k
0 votes
0 answers
144 views

Signed distance function

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function: $d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
  • 1,092
2 votes
1 answer
152 views

What are conditions to satisfied by rational prime p so that every prime lying above p is a prime of order 1 and generates class group?

I was reading a paper on Euclidean ideals by H Graves and M. Ram Murthy. I have a problem in understanding one of the claims. setup Let $K$ be a number field and $H(K)$ is its Hilbert class field. ...
2 votes
0 answers
124 views

Limit involving singular kernel: $\lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y. $

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \...
  • 763
1 vote
0 answers
59 views

Regularity of a shrunken domain

I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner. Let $\Omega\subset\Bbb R^d$ be an open bounded (...
  • 763
0 votes
0 answers
55 views

Is there an analogy to twin primes in the rational integers among the Gaussian and Eisenstein integers?

Twin primes, like (29, 31) and (137, 139) are interesting to study. I have been exploring the parallels of the Gaussian and Eisenstein integers with the rational integers. For instance, they have ...
user avatar
6 votes
0 answers
137 views

Quadratic function fields that are norm-euclidean or PIDs

It is known that there are finitely many square-free integers $d$ for which the ring of integers of $\mathbf{Q}(\sqrt{d})$ is Euclidean under the norm function. Is there an analogous result for ...
  • 81
4 votes
0 answers
97 views

A Euclidean domain in which radix expansion is not possible

It is well known that given two nonconstant polynomials $f,g\in F[x]$ where $F$ is a field, there are unique polynomials $r_0,\dots ,r_n$ such that $$f=r_n g^n +\dots+r_1 g +r_0,$$ where $\deg r_i &...
2 votes
1 answer
253 views

About Euclidean domains

I asked a similar question a few weeks ago in M.SE but it didn't receive any answers, so I decided to post it here with some modifications. My motivation comes from a theorem given in Pete L. Clark's ...
  • 195
11 votes
1 answer
4k views

How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that $n = \sum_{i=1}^k a_i m_i$? ...
  • 231
4 votes
0 answers
325 views

Comparing different Euclidean algorithms on a Euclidean domain

I have posted this question at stackexchange (502413), without responses until now. In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...
20 votes
4 answers
8k views

For which $c$ is $\mathbb{Z}[\sqrt{c}]$ a unique factorization domain? a Euclidean domain?

Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, \quad a, b\in \mathbb{Z},$$ which form a subring of the ...
  • 724
9 votes
1 answer
2k views

Reference request: number theory of Z[1/p]

Can anyone suggest a good place to read up on the number theoretic properties of and techniques for $\mathbb{Z}[1/p]$, (that is, rational numbers with only powers of a prime $p$ in the denominator)? ...
  • 2,155
22 votes
3 answers
1k views

Must a ring which admits a Euclidean quadratic form be Euclidean?

The question is in the title, but employs some private terminology, so I had better explain. Let $R$ be an integral domain with fraction field $K$, and write $R^{\bullet}$ for $R \setminus \{0\}$. ...
43 votes
2 answers
4k views

Why do we care whether a PID admits some crazy Euclidean norm?

An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > 0$,...