-1
votes
1answer
80 views

derivatives and uniformly convergence [closed]

Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has ...
3
votes
1answer
96 views

Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions: (i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$; (ii) $f$ is non-degenerate, in the sense that there isn't a ...
0
votes
1answer
95 views

The Largest Root of Associated Laguerre Polynomial

The Laguerre polynomial $L_n(x)$ is the solution to the Laguerre differential equation \begin{equation*} x\,y'' + (1 - x)\,y' + n\,y = 0. \end{equation*} The associated Laguerre polynomial ...
0
votes
1answer
88 views

k-th largest root in common interlacing polynomials

In their proof of the celebrated Kadison-Singer conjecture, Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on Ramanujan graphs. ...
3
votes
1answer
65 views

Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$. I would like to know how many extrema $p$ has on the standard simplex ...
9
votes
1answer
527 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
3answers
505 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 ...
1
vote
2answers
260 views

Bounds on the largest root of a polynomial

Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x_{\max}$ of the ...
2
votes
2answers
155 views

Finding a simpler “local” lower bound for a rational function

I have obtained as the expression for some quantity the following gargantuan formula: $$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$. ...
1
vote
1answer
134 views

Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
6
votes
0answers
289 views

Polynomials and divided differences

I would greatly appreciate any hint for proving the following. Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then ...
9
votes
2answers
594 views

When polynomial f(x^2) can be factored as g(x)·g(-x) ?

In relation to my question Expression for the sum of square roots of zeros of a polynomial How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where ...
5
votes
2answers
427 views

Expression for the sum of square roots of zeros of a polynomial

Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$. General question. Does there exist a simple expression for the ...
1
vote
0answers
331 views

Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/. Assume that all constants and exponents are integers within certain range. I ...
3
votes
3answers
362 views

Every positive polynomial with rational coefficients is above a completely Q-factorized nonnegative polynomial ?

This question was originally asked in stackoverflow (http://math.stackexchange.com/questions/103941/every-positive-polynomial-is-above-a-completely-q-factorized-positive-polynomial) but as it has ...
5
votes
2answers
552 views

Stone-Weierstrass for monotone functions

Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 < \epsilon \;$. Does it follow that that there exists a real ...
13
votes
2answers
901 views

Constants for Rolle's Theorem applied to polynomials

Rolle's Theorem states that $f(1/2)=f(-1/2)+f'(x)$ has a root in the open real interval $(-1/2,1/2)$ if $f$ is continuous and differentiable. How large can the absolute value of such a root $\xi$ be ...