# Tagged Questions

**1**

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**0**answers

129 views

### Solving polynomials of arbitrary degree

Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5?
More specifically: does there exist ...

**3**

votes

**1**answer

125 views

### Generalization of the Hermite-Bielher-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...

**1**

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**0**answers

75 views

### Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...

**4**

votes

**2**answers

212 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

534 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 ...

**10**

votes

**3**answers

512 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

88 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 ...

**3**

votes

**1**answer

391 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**

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**0**answers

436 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**

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**2**answers

217 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 ...

**12**

votes

**4**answers

770 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**

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**0**answers

144 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**

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**0**answers

170 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

392 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

246 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 ...

**0**

votes

**1**answer

695 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

193 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

197 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

223 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**

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**0**answers

236 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

412 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

211 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

429 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

452 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

355 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

334 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 ...

**13**

votes

**0**answers

630 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 ...

**9**

votes

**1**answer

866 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 ...

**11**

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

603 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

507 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

525 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**

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**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 ...