Questions tagged [euclidean-domain]

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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

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 $E$ ...
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9 votes
1 answer

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

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

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

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

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

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

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

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 ...
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6 votes
0 answers

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 ...
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4 votes
0 answers

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

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 ...
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11 votes
1 answer

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 $n = \sum_{i=1}^k a_i m_i$? ...
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4 votes
0 answers

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

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

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

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

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 $N(y) > 0$,...