# Tagged Questions

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### Gaps between roots of trigonometric polynomials

[Cross-posted from Math.SE because I got no responses there.] Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t}$$ with $\bar c_{-k} = c_k$, ...
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### Roots of modified polynomials

Consider the following two polynomials: $$g=x^3 - x^2 - (c + 2)x + c$$ and $$h=x^3 - x^2 - cx + c$$ The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...
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### is there any such result about Bernstein polynomials?

It is well known that for any lipschitz function $f:[0,1]\rightarrow [0,1]$, we can approximate it by $\sum_{i=1}^n f(i/n) {n\choose i} x^i (1-x)^{n-i}$, and the $L_\infty$ error is $O(1/\sqrt{n})$. ...
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### unique positive real root fast computation

What is the fastest way to compute the value of the unique positive real root corresponding to the following polynom: :p(x) = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x - f = 0 where a, b, c, d, e, f are ...
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### Do interpolation nodes have to be dense?

Let $f(x) = \exp(x)$ and $(\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1)$ be a sequence of points from the unit interval. For $n \in \mathbb{N}$ let $P_n$ be a polynomial of degree $n$ that interpolates ...
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### Solving a System of Quadratic Equations

I have many polynomial equations in many variables which I want to jointly minimize (in a mean square sense, but you could pick a different reasonable measure which favors anything where all ...
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### Is there a quick way to find all roots of a real polynomial with multiple variables?

If I am asked to find the roots of a polynomial of one variable, I will use a computer to estimate the eigenvalues of its companion matrix. Now suppose I'm given a real polynomial of multiple ...
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### Finding a low-degree polynomial vanishing on half the zeroes of a polynomial system

Let $f(x)$ be a real polynomial of degree $2d$ without real roots. Let the complex roots be $z_1$, $\bar{z_1}$, $z_2$, $\bar{z_2}$, ..., $z_d$, $\bar{z_d}$ with $z_i$ in the upper half plane. Let ...
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### Value of coefficient in estimation of computational complexity of polynomial division algorithm

Do you know value of coefficient $C$ at $C*n*log(n)$ in $O(n*log(n))$ estimation of complexity of polynomial division algorithm? It would be great if you give me links to paper with information about ...