7
votes
1answer
158 views

Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by $\quad L_n (x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad $ I want to prove that $$ \forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...
0
votes
0answers
82 views

Lower-Upper bounds on the cardinality of a set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...
7
votes
0answers
449 views

Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$, \begin{cases} ...
4
votes
1answer
342 views

Inequality on Trigonometric polynomials

My question comes from trying to understand a technical step in this paper by Bourgain. Let $R,L$ be positive integers and let $f(x)=\sum_{|n|\leq RL}a_ne^{2\pi inx}$ be a trigonometric polynomial. ...
5
votes
1answer
213 views

Convexity of a specific semialgebraic set

I have an engineering problem which maybe resolved with semi-definite programming optimization. I have a set which I would like to know if is convex: Being $m \in \mathbb{R}^+$ a positive real ...
1
vote
1answer
190 views

Mapping multivariate polynomial inequalities system to subspace

What I will ask, more than a solution, is better mathematical definition of my problem and directions to find the solution. I have a set of linear equations, e.g.: \begin{align} d_1 &= L_1 - ...
2
votes
3answers
387 views

About Turan`s problem(inequality) in multivariable

Hi. I have a question related to Turan`s problem, that is Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$. I am considering the generalized question for ...
3
votes
2answers
408 views

Positivity of a finite sum

Let $i$, $k$ be integers such that $2 \leq i \leq k$. I would like to show that the sum $$ \sum_{j=1}^{i-1} \frac{(-1)^{j-1}(i-j)^k}{(i-j)! (j-1)!} $$ is positive. I have carried out extensive ...
5
votes
3answers
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 ...