All Questions
10 questions
4
votes
0
answers
179
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As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?
As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
3
votes
0
answers
226
views
On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial
Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
- 429
0
votes
0
answers
109
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The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool
In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
2
votes
1
answer
120
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Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$
Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$.
Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of
$$p(w)=n+\sum_{j=1}^{m}\frac{...
- 5,474
7
votes
1
answer
244
views
Volume of solution sets for polynomials in $\mathbb{C}[x]$
Denote $\pmb{a}=(a_1,\dots,a_d)\in\mathbb{R}^d$ and consider the set
$$\mathcal{E}_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a_dx^{d-1}+\cdots+a_2x+a_1=0$ lies in $\vert\xi\vert<1$}\...
- 43.1k
2
votes
0
answers
180
views
Multiple zeta values related to fractional calculus and an Appell polynomial sequence
There is an Appell sequence of polynomials $p_n(z)$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at ...
- 10.5k
4
votes
1
answer
268
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On the roots of Bernoulli polynomials
Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...
- 43.1k
4
votes
3
answers
667
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Regularity for the roots of (characteristic) polynomials with given multiplicity
A classical result states that roots of a polynomial are continuous functions of its coefficients.
This is, for exemple, a direct consequence of Rouché's theorem.
Using the implicit function ...
- 2,135
6
votes
0
answers
332
views
Criteria for irreducibility using the location of complex roots
I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
- 11.3k
11
votes
3
answers
899
views
How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?
This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch.
EDIT:
This is an edited version. Before I asked about roots ...
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