25
votes
Accepted
Irreducibility of the polynomial $x^n+5x+3$ over $\mathbb{Q}$
This should be provable using the results of On the irreducibility of the non-reciprocal part of polynomials of the form $f(x) x^n+g(x)$ by Filaseta-Li-Patane-Skabelund Acta Arithmetica 196 (2020), ...
- 137k
11
votes
Accepted
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
If you plot $\log f_n$ versus $n$, with $f_n=\sum_i x_i^{2n}$, then the asymptotic slope for large $n$ will give you the largest of the $x_i$; subtracting that contribution from $f_n$ and repeating ...
- 178k
9
votes
Accepted
About nonnegative polynomials
This is a simple linear programming problem, which has no solutions.
Here are the calculations, in Mathematica:
- 118k
8
votes
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
If $\sum x_i^2$ is finite, the sum $f(z)=\sum \frac{x_i^2}{1-x_i^2z}$ is a meromorphic function on the complex plane, and we know its Taylor series at 0. Thus we know $f$, hence the poles of $f$, ...
- 103k
7
votes
Accepted
Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices
We can show that for every $m$ there is a polynomial $q_m$ of degree $\binom m2$ such that $G_{m, n} = (m+1)^n q_m(n)$. We will do this by constructing a matrix $A_m$ such that $G_{m, n} = \mathbf{1} ...
- 2,979
6
votes
Accepted
A version of Hilbert's Nullstellensatz for real zeros
I think, it does.
By change of coordinates, you may suppose that $Z$ contains the origin and the tangent vector space is the hyperplane $\{x_n=0\}$. Then, by implicit function theorem, for small ...
- 103k
5
votes
Accepted
Can the Chebyshev polynomials be constructed from the extremal property?
The argument in the answer you refer to actually shows that (using the notation from that answer) $t_2, \ldots, t_n$ are extrema of $p$ with value $\pm 1$ and thus double roots of $1-p^2$. Hence $(1-x^...
3
votes
Accepted
On properties of sums involving the floor function
We first note that $f(n,k)=1-\frac{(2^n+1)\%(2^k+1)}{2^n+1}$. Therefore
$$\sum_{k=1}^{n}\left(1-f(n,k)\right)k^m = \sum_{k=1}^{n-1}\frac{(2^n+1)\%(2^k+1)}{2^n+1}k^m,$$
we can replace the denominators ...
- 186
3
votes
When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?
To make this question precise, yo have to specify the analytic nature of solutions that you consider: where they are defined/analytic. When speaking of doubly periodic solutions, they usually require ...
- 88.9k
2
votes
Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices
Command Master has already answered the question nicely. For my own understanding, this answer works out the details of that answer in the $m=3$ case (excluding the linear algebra at the end). Here's ...
- 490
2
votes
Proofs of the Chevalley-Warning Theorem
Whether the following proof is different from the other ones already mentioned could be debated, but to me it feels different enough to be worth mentioning separately. As noted in Gjergji Zaimi's ...
- 78.6k
2
votes
On zeros of real polynomials in two variables
If all zeros of $Q$ in $\mathbb C^2$ are zeros of $P$, then, by Hilbert's Nullstellensatz, $Q$ divides $P^r$ for some natural $r$.
If $Q$ is also irreducible, then it follows that $Q$ divides $P$.
If $...
- 118k
2
votes
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
In terms of the finite and compactly supported measure $\mu=\sum x_j^2 \delta_{x_j^2}$, we are given the moments $\int t^k\, d\mu(t)$, $k=0,1,2 \ldots$.
Moment problems with compactly supported ...
2
votes
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
[edit: completed] Assuming $x_i\ge0$ with $ \sum_i x_i <\infty$, we have that $\phi(t):=\sum_i(e^{x_it}-1)=\sum_{k\ge1} \big(\sum_i x_i^k\big)t^k/k!$ is an entire function (we can expand the ...
- 56.6k
2
votes
A version of Hilbert's Nullstellensatz for real zeros
The argument from the link that OP included can be mimicked: reducing the problem to complex Nullstellensatz with a bit of analysis. Consider a non-singlular point $\mathbf{p}\in Z$ near which the ...
- 2,817
1
vote
Abstract algebraic link between two problems involving polynomials and (generalized) Vandermonde matrices?
Well, after some more thinking I'm going to answer my own question. It was pretty much just a matter of linking all elements together.
Here are my notations, in $\mathbb R_N[X]$.
Note $\partial$ the ...
1
vote
Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$
Here is what happens without assumptions.
Functions $p_n:=\sum_i x_i^n$ represent power-sum symmetric polynomials. By Newton's identities, we have
$$E(t):=\sum_{k=0}^\infty e_k \,t^k = \exp\left(\sum_{...
- 30.7k
1
vote
Accepted
Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?
This answer is a slight revision of the proof of Proposition 1 in Section 4 of the forthcoming paper:
Gui-Zhi Zhang, Zhen-Hang Yang, and Feng Qi, On normalized
tails of series expansion of generating ...
- 838
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