# Tagged Questions

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 ... 0answers 79 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} ... 0answers 440 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} ... 1answer 322 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. ... 1answer 209 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 ... 1answer 176 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 - ... 2answers 367 views ### About Turans problem(inequality) in multivariable Hi. I have a question related to Turans 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 ... 2answers 402 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 ...
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 ...