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6 votes
2 answers
492 views

Does this polynomial have a real zero less than or equal to $1/2$?

Is the smallest root $x$ of $$ 10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\ +2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
9 votes
3 answers
2k views

Smallest root of a degree 3 polynomial

Is it true that the smallest root $t$ of the polynomial $$ 20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
3 votes
1 answer
240 views

A certain generalisation of the golden ratio

Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$ We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...
11 votes
3 answers
4k 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 ...
31 votes
1 answer
1k views

Finding a path through real rooted polynomials

This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true. Let $z^n+a_{n-1} z^{n-1} + \cdots + ...
7 votes
1 answer
341 views

L1 analog of Bernstein's inequality

Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that $$ \|q\|_1 \leq O(n) \|p\|_1 $$ where we define $\|f\|_p := \left(\int_{-1}^1 |f(x)|^pdx\...