All Questions
12 questions
17
votes
1
answer
502
views
Irreducibility of root-height generating polynomial
The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of ...
- 2,753
67
votes
6
answers
7k
views
How to recognise that the polynomial method might work
A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let $(a_1,b_1),\dots,(...
- 29k
37
votes
2
answers
3k
views
A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself
Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
- 1,730
6
votes
1
answer
521
views
Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
Is it true that for each irreducible ...
- 11.3k
26
votes
0
answers
910
views
Which sets of roots of unity give a polynomial with nonnegative coefficients?
The question in brief: When does a subset $S$ of the complex $n$th roots of unity have the property that
$$\prod_{\alpha\, \in \,S} (z-\alpha)$$
gives a polynomial in $\mathbb R[z]$ with ...
- 1,513
14
votes
2
answers
1k
views
Number of nonzero terms in polynomial expansion (lower bounds)
Let $f(x) = a_1x^{z_1} + a_2x^{z_2} + \cdots + a_kx^{z_k}$ be a polynomial with coefficients $(a_1, \ldots, a_k) \in \mathbb{F}_q^*$ and $z_i$ are distinct positive integers. If I need to compute the ...
- 165
9
votes
1
answer
1k
views
"Laurent phenomenon"?
Define the recurrence
\begin{align*}
n(2n+x-3)u_n(x)
&=2(2n+x-2)(4n^2+4nx-8n-3x+3)u_{n-1}(x) \\
&-4(n+x-2)(2n-3)(2n+2x-3)(2n+x-1)u_{n-2}(x)
\end{align*}
with initial conditions $u_0(x)=0$ and $...
- 43.2k
9
votes
2
answers
538
views
Counting monomials in product polynomials: Part I
This question is motivated by recent work of R P Stanley, Theorems and conjectures on some rational generating functions. Consider the polynomials
$$P_n(x)=\prod_{i=1}^{n-1}(1+x^{3^{i-1}}+x^{3^i}).$$
...
- 43.2k
3
votes
1
answer
447
views
A number array related to colored necklaces and the primes
I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
- 10.5k
3
votes
2
answers
405
views
Irreducible Polynomials from a Reccurence
This question is inspired by a recent one : Let $c$ be a variable and define a sequence by $a_0=0$ $a_1=1$ and $a_{n+1}=a_{n}c-a_{n-1}$ . So
$$\begin{align*}
a_2 &= c
\\ a_3 &={c}^{2}-1= \...
- 30.1k
3
votes
0
answers
144
views
Flat polynomials with factors of big height
Let $p(x)$ be a polynomial of degree $n$ with all coefficients in $\{-1,0,1\}$ (such polynomials are sometimes called flat). I am wondering how big the coefficients of a factor of $p$ can be. Call ...
- 13.4k
2
votes
1
answer
385
views
Determinants of striped Hankel matrices
This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
- 43.2k