All Questions
12 questions
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 1,889
13
votes
2
answers
539
views
$f$ real-rooted forbid truncated $\frac1f$ to be so?
Let $f(x)$ be a polynomial in the ring $\mathbb{R}[x]$, the roots are all real and $f(0)=1$. Write the Taylor series of $1/f(x)$ around the origin as
$$\frac1{f(x)}=\sum_{k=0}^{\infty}a_kx^k,$$
and ...
- 43.1k
6
votes
4
answers
780
views
roots of higher derivatives of exponential
Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$.
Question. Is it true that $D^nf(z)=0$ has only real roots ...
- 43.1k
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
4
votes
3
answers
667
views
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
3
votes
1
answer
137
views
Estimate the homogeneous components of a polynomial against its maximum
Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.
(I.e., the above sum ranges over ...
- 463
2
votes
1
answer
115
views
Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane
Suppose that
$f$ and $g$ are polynomials with nonnegative coefficients,
the degree of $g$ is greater than the degree of $f$,
$g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...
- 187
1
vote
2
answers
151
views
Location of the negative real roots of certain integer-valued polynomials
The following question on polynomials arose as a potentially helpful intermediate step on a proof of a Theorem that I want to demonstrate. Its statement is quite elementary, and I can think of a ...
- 1,889
1
vote
2
answers
588
views
Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
0
votes
1
answer
116
views
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots.
Based on an old paper (this reference), it has been ...
- 4,058
0
votes
1
answer
662
views
A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
- 309
0
votes
1
answer
152
views
When can two Cauchy transforms intersect?
Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...
- 1,893