All Questions

$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that $$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...

For each real $A>0$, let $x_A$ denote the positive root $x$ of the polynomial $x^5-3x-A$. Is the function $(0,\infty)\ni A\mapsto x_A$ elementary? [I am using this definition of elementary ...

Dyson's Theorem The constant term in the expansion of $$\prod_{1\leq i\neq j\leq n}\left(1-\frac{x_i}{x_j}\right)^{a_i}$$ is the multinomial coefficient $$\frac{(a_1+\cdots+a_n)!}{a_1!\cdots a_n!},$$ ...

Let \begin{align} P(x) &= x^n+a_{n-1} x^{n-1} +\ldots+a_0 = \prod_{i=1}^n (x-p_i)\\ Q(x) &= x^n + b_{n-1} x^{n-1} + \ldots +b_0 = \prod_{i=1}^n (x-q_i)\\ p_i, q_i& \in \mathbb{R},\ 0<...

Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables? In the 2-dimensional case the ...

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 ...