# Tagged Questions

**4**

votes

**2**answers

157 views

### Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...

**9**

votes

**1**answer

478 views

### Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence
\begin{equation}
a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),
\end{equation}
with $a(1,1)=1$ and ...

**9**

votes

**3**answers

487 views

### Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...

**7**

votes

**2**answers

82 views

### Bounds on coefficients of factors of a multivariate polynomial

Given a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that can be quickly found such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote the ...

**0**

votes

**0**answers

90 views

### Bound on a sum of Laguerre polynomials

I am trying to find an asymptotic behavior, for large real $t$, of the following sum
\begin{align}
Q(t)=\sum_{0\le n\le t}e^{-(t-n)}\frac{t-n}{1+n}L_n^{(1)}(t-n)
\end{align}
where $L_n^{(\alpha)}$ is ...

**3**

votes

**1**answer

336 views

### Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...

**17**

votes

**0**answers

363 views

### What is the best probabilistic estimate from below for a random polynomial on an arc?

I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k ...

**1**

vote

**2**answers

200 views

### Approximation by polynom 1) with respect to supremum-norm 2) I need F_{approx} > F_{exact}

Given a function F, how to find polynom which is best/good approximate with respect supreremum-norm, i.e. minimize over P_{approx} sup|F-P_{approx}| ?
I am intersted in polynoms in two variables of ...

**11**

votes

**4**answers

641 views

### Showing that a family of polynomials has positive and real roots.

Hi everybody, for my research I am dealing with the following function:
$$\alpha_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|_{z=0},\quad n\in ...

**2**

votes

**0**answers

141 views

### Growth of the recurrence coefficients of orthogonal polynomials

Consider the sequence of measures $$d\mu_N(x)=e^{-NV(x)}dx$$ on the real axis, where $V$ is continuous and satisfies the growth assumption $$\lim_{|x|\rightarrow\infty}(V(x)-2\log|x|)=+\infty.$$
...

**2**

votes

**0**answers

163 views

### Markov-type inequalities with arbitrary exponents

By a Markov-type inequality I mean an inequality of the form
$$
\| p^{(k)} \| \leq \lambda_{k,n} \| p \|,\quad \forall p \in U_n,
$$
for some $\lambda_{k,n} > 0$, where $U_n \subset L^\infty[-1,1]$ ...

**5**

votes

**1**answer

387 views

### Polynomials with prescribed points to match prescribed bounds

Consider real polynomials on the interval $I=[-1,1]$. It is easy
to see that the smallest degree for a non-negative polynomial
with given zeros $x_1,\dots,x_s\in I^\circ$ is $n=2s$ (e.g.
$P(x) = ...

**1**

vote

**1**answer

245 views

### Extension of polynomial functions

Let $P$ be an interval in $\mathbf{R}$, $n \in \mathbf{N}$. Assume that a function $f: P \rightarrow \mathbf{R}$ satisfies $\Delta^{n+1}_h f(x)=0$ for every $x \in P$ and every $h>0$ such that ...

**1**

vote

**1**answer

692 views

### A nice problem that I am unable to solve [closed]

I have a nice problem which is related to algebra and polynomials or even operator theory. I would be very grateful if any of you could solve it. Here is the problem: Consider the function $f_0(x) ...

**2**

votes

**4**answers

1k views

### Positive polynomials

How are polynomials called that are positive for all positive, real arguments, e.g., xy + z?
How can one determine if a polynomial has this property?

**1**

vote

**1**answer

192 views

### evaluation for homogeneous polynomials

Let $p:=\sum_{n=0}^\infty p_n$ be a polynomial given by its terminating decomposition by means of homogeneous polynomials. For fixed $x\in \mathbb{R}^d $ and an none negative integer $n$, can we find ...

**1**

vote

**0**answers

192 views

### exponential scaling and polynomials

Consider the following statement $S_a$, with some parameter $a > 1$ :
"If $P(X)$ is a real polynomial such that for every $i\in \{0;1;\ldots;N\}$, $|P(a^i)|< 1$ and $|P(a^0)-P(a^1)|>1$, then ...

**0**

votes

**1**answer

220 views

### algebraic vis-a-vis analytic functions

Consider functions $\mathbb{R}^n\to\mathbb{R}$. If the set of algebraic functions is the algebraic closure of the rationals, does this mean that limits of polynomials are algebraic? In particular, are ...

**9**

votes

**0**answers

226 views

### Positivity of polynomial sequences via generating series

In this question I address
the problem of proving the nonnegativity of a numerical sequence
$a_0,a_1,a_2,\dots$ via generating series technique. In the notation
$A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...

**2**

votes

**1**answer

406 views

### Is every polynomial a limit of polynomials in quadratic variables?

Let $d>0$ be even. Consider ${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. polynomials of degree $d$.
Call a homogeneous polynomial $f$ of degree $d$ a polynomial in quadratic variables if it is of the ...

**5**

votes

**0**answers

207 views

### Polynomial upper approximation with respect to the Gaussian measure

Let $f = 1_{[a,+\infty)}$ be the indicator function of a half-line. Does there exist a sequence $(P_n)$ of polynomials such that $f(x) \leq P_n(x)$ for every real $x$ and
$$ \lim_{n\to \infty} ...

**6**

votes

**2**answers

2k views

### Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb ...

**9**

votes

**3**answers

426 views

### Effective algorithm to test positivity

Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?

**0**

votes

**1**answer

450 views

### modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder
$R_{k,m} \equiv H_{k} ~ \mod H_m$
for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial ...

**2**

votes

**1**answer

1k views

### Puiseux series for roots of polynomials with smooth coefficients

If
$$p(x,y) = x^N + a_{N-1}(y)x^{N-1} + \ldots + a_0(y), \quad x,y \in \mathbf{C}$$
is a monic polynomial in $x$, and the coefficients $a_j$ are analytic functions of $y$, then the roots of $p$ ...

**1**

vote

**2**answers

349 views

### Finding regions where multi-variate polynomials are positive

Given a constant $k \in \mathbb N$, and a set of $p$ multi-variate polynomials {$P_j:\mathbb N^n\to \mathbb Z$}$\_{j=1...p}$, with $P_j \not\equiv 0$.
Is the following true:
There exists $n$ sets ...

**6**

votes

**2**answers

330 views

### How large (small) can be the measure of a set where a polynomial takes small values ?

A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...

**12**

votes

**0**answers

603 views

### Polynomials with presumably positive coefficients

After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...

**20**

votes

**1**answer

2k views

### Polynomials with rational coefficients

Long time ago there was a question
on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt
of answering it has been given, highly downvoted by the way. But this answer ...

**16**

votes

**3**answers

1k views

### Recursions which define polynomials

There are many examples (Somos sequences, special polynomials related to rational solutions of the Painleve equations) when a recurrence relation, which a priori produces a sequence of rational ...

**8**

votes

**1**answer

850 views

### How to best distribute points on two concentric circles?

An N-subset $\{x_1,\dots,x_N\}$ of a compact set $X\subset \mathbb R^d$ is called a set of Fekete points (named after Michael Fekete) if it maximizes the product $$\prod_{1\le k<j\le ...

**10**

votes

**5**answers

7k views

### Criteria to determine whether a real-coefficient polynomial has real root?

Given a polynomial equation $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is even and all the coefficients $a_i$ are real, what is the best way to determine whether it has a real root or not?
I ...

**11**

votes

**2**answers

1k views

### Growth of the “cube of square root” function

Hello all, this question is a variant (and probably a more difficult one)
of a (promptly answered ) question that I asked here, at Is it true that all the "irrational power" functions are ...

**4**

votes

**5**answers

565 views

### Asymptotic series for roots of polynomials

Let $f(z) = z + z^2 + z^3$. Then for large $n$, $f(z) = n$ has a real solution near $n^{1/3}$, which we call $r(n)$. This appears to have an asymptotic series in descending powers of $n^{1/3}$, ...

**3**

votes

**3**answers

500 views

### Polynomials and L^p(R)

As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...

**5**

votes

**3**answers

2k views

### When does a real polynomial have a pair of complex conjugate roots?

Suppose we have a polynomial function $f(z)=a_0+a_1z+a_2z^2+...+z^n$ with each $a_i$ between 0 and 1. Is there a method to determine if $f$ has a pair of complex conjugate roots?
There are many ...

**8**

votes

**1**answer

513 views

### what was Hilbert's geometric construction in his 17th problem?

Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...

**18**

votes

**11**answers

3k views

### When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...