Unanswered Questions
7,519 questions with no upvoted or accepted answers
21
votes
0
answers
430
views
Straight-line drawing of regular polyhedra
Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...
- 4,282
21
votes
0
answers
597
views
Coloring a Ferrers diagram
I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed.
Say that a coloring of the dots of a Ferrers ...
- 80.3k
21
votes
0
answers
606
views
Bounding failures of the integral Hodge and Tate conjectures
It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
- 20.9k
21
votes
0
answers
2k
views
The Fourier Transform of taking Eigenvalues
The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
- 51.9k
21
votes
0
answers
902
views
Cauchy matrices with elementary symmetric polynomials
$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 \...
- 28.5k
21
votes
0
answers
2k
views
K-theory and rings of integers
From the works of Borel and Quillen there is a connection between the $K$-theory of the ring of integers $\mathfrak{o}_K$ in a number field $K$ and the arithmetic of the number field. In fact, it is ...
- 1,831
21
votes
0
answers
894
views
Deciding whether a given power series is modular or not
The degree 3 modular equation for the Jacobi modular invariant
$$
\lambda(q)=\biggl(\frac{\sum_{n\in\mathbb Z}q^{(n+1/2)^2}}{\sum_{n\in\mathbb Z}q^{n^2}}\biggr)^4
$$
is given by
$$
(\alpha^2+\beta^2+6\...
- 13.4k
21
votes
1
answer
1k
views
Tiling rectangle with trominoes — an invariant
There are two types of trominoes, straight shapes and L-shaped. Suppose a rectangle $R$ admits at least one tiling using trominoes, with an even number of L-trominoes.
EDIT: we do not admit ALL ...
CommunityBot
- 1
20
votes
0
answers
481
views
Is every positive integer the rank of an elliptic curve over some number field?
For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$?
It's easy to show that the set of such $n$ is unbounded. But can one show that ...
- 15.5k
20
votes
0
answers
964
views
Finiteness of etale cohomology for arithmetic schemes
By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$.
Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...
- 20.9k
20
votes
0
answers
887
views
Looking for an effective irrationality measure of $\pi$
Most standard summaries of the literature on irrationality measure simply say, e.g., that
$$
\left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}}
$$
for all sufficiently large $q$, without giving ...
CommunityBot
- 1
20
votes
0
answers
559
views
Hall's Marriage Theorem and intervals
In Hall's Marriage Theorem, we have a set $B$ of brides and $G$ of
grooms, where each bride $b$ has an acceptable set $A_b \subseteq G$
of grooms. A matching $m:B\to G$ is an injection such that $m(b)
...
- 27.7k
20
votes
0
answers
1k
views
Could unramified Galois groups satisfy a version of property tau?
This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...
- 19.1k
19
votes
0
answers
595
views
Large values of characters of the symmetric group
For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the ...
- 1
19
votes
0
answers
420
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Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
- 23.1k