All Questions
12 questions
17
votes
1
answer
687
views
Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I have copied this question from StackExchange, in the hope that some experts here can provide some relevant insight. Thanks to Greg Martin for improving the question.
Given $f(x) = a_0 + a_1 x + a_2 ...
- 7,893
1
vote
0
answers
162
views
Difficulty understanding a step in the proof of multiset version of Cauchy-Davenport Theorem
In a paper "G. Kós, L. Rónyai, Alon’s Nullstellensatz for multisets, Combinatorica, 32(5) (2012) 589-605", the authors prove a multiset version of the Cauchy-Davenport Theorem (please see ...
- 167
0
votes
0
answers
132
views
Which consequences can be deduced from this peculiar property of tetration?
Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of ...
- 1,451
6
votes
1
answer
242
views
$(q,t)$-Fibonacci polynomials: area & bounce statistics
This is related to my earlier (unanswered) MO post. Preserve notations from there.
We take advantage of the one-to-one correspondence between the $(s,s+1)$-core partitions and $(s,s+1)$-Dyck paths. ...
- 85.6k
7
votes
1
answer
474
views
Fibonacci embedded in Catalan?
Given a partition $\lambda$ and its Young diagram $\pmb{Y}_{\lambda}$, we say $\lambda$ is a $(t,s)$-core partition provided that neither $t$ nor $s$ is a hook length in $\pmb{Y}_{\lambda}$. We now ...
- 85.6k
18
votes
5
answers
2k
views
Is a complete homogeneous symmetric polynomial irreducible?
Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$.
$$ h_a=\text{ sum of all monomials of degree }...
2
votes
1
answer
161
views
How to classify solutions to the following equations?
I would like to classify the sets of integers $a_{1},...,a_{n}$ that satisfy the following two equations.
$$\sum_{k=1}^{n}a_{k}\equiv 0\mod 2$$
$$\sum_{i\neq j}a_{i}a_{j}=0$$
For example, if $n=3$, I ...
- 54.2k
1
vote
0
answers
164
views
Combinatorial splitting in number rings
The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring.
Take an arbitrary non empty ...
CommunityBot
- 1
4
votes
1
answer
457
views
Circulant matrix with integer entries and determinant 1 or -1
CONJECTURE
Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$th column of $A$ is obtained by applying the permutation $(1,2,..,n)...
CommunityBot
- 1
1
vote
2
answers
509
views
An extremal combinatorics problem over Finite Rings
Cross Posting from: https://math.stackexchange.com/questions/462016/a-combinatorics-problem-over-finite-rings
Consider the set $S$ of all non-zero vectors over $\Bbb Z_{q}$ of length $r$ whose ...
CommunityBot
- 1
2
votes
1
answer
255
views
What does the d-slice of a weighted polynomial algebra look like?
This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves.
...
- 27.8k
3
votes
7
answers
4k
views
How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
- 32.1k