# Questions tagged [roots]

Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots" and such, consider using the (radicals) and the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

751 questions
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### How to prove this lemma related to Rolle's theorem

For any function $f$ denote by $Z(f)$ and $Z_o(f)$ the cardinalities of $f^{-1}(0)\cap[0,1]$ and $f^{-1}(0)\cap(0,1)$, respectively. Let $H=\{f\in C^\infty(\mathbb{R}): \text{supp}(f) = [0,1]\}$ From ...
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### The Heegner Polynomials

What is special about $x^3- 6 x^2 + 4 x -2$? The 24th power of the real root - 24 is curiously close to two other numbers, one being the Ramanujan constant. There are more of these polynomials ...
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### Generating Functions, Recursive Polynomials

At the CMFT international conference in Turkey (2009), the following open problem was given: Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
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### Finding zeros of function by integration: a novel relationship or not?

It seems that in certain cases one can find the zero of a function by solving an integration problem instead. This surprises me, and I am wondering to what extent this (1) has been studied, and/or (2) ...
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### Let $\lambda_n$ be the location of the $n^{th}$ positive root of $x = 1/\sin(x)$. Evaluate $\sum_{n=0}^{\infty}\left(\lambda_n - n\pi\right)$

Consider the set of positive intersections between the functions $x$ and $1/\sin(x)$. These are shown as red dots in the image below: Let $\lambda_n > 0$ be the $x$-coordinate of the $n^{th}$ such ...
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### Regarding recurrences, why do characteristic polynomials work, and why do we look for the roots?

I'll use an example recurrence but my question is meant to be generalized. Let's say we had some recurrence, such as: $$F(n) = -8F(n-1) + 9F(n-2) + 92F(n-3) - 140F(n-4)$$ where we already know the ...
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### solution set in $\mathbb{C}$ of $z^{\frac1{z}}=\left(\frac1{z}\right)^z$

If $z \in \mathbb{C}$ what can be said about the solution set of: $$z^{\frac1{z}}=\left(\frac1{z}\right)^z$$ aside from the fact that it contains the fourth roots of unity? I will add as a footnote ...
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### Determine the number of zero points of $z^8-5z^3+z-2$ within the open unit circle (Rouché?)

How many zero points does the polynomial $z^8-5z^3+z-2$ have within the open unit circle? Hello, consider $$\gamma\colon [0,2\pi]\to\mathbb{C}, \varphi\longmapsto\exp(i\varphi)$$ and define ...
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### Methods of solving roots involving matrix determinant

I have a square matrix $F$ whose elements depend non-linearly on a complex parameter $s$. I would like to know the values of $s$ such that $\det(F)=0$, i.e., those $s$ that make $F$ singular. I would ...
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### Borsuk–Ulam theorem proof using Brouwer degree

I wonder if Borsuk–Ulam theorem (if $f:\mathbb{S}^n\rightarrow\mathbb{R}^n$ is continuous, then exists $x_0\in\mathbb{S}^n$ such that $f(x_0)=f(-x_0)$) can be sucesfully proved by using the Brouwer ...
I have the function $$f (x) = \cos x - \frac{x^2}{100}$$ $$f'(x) = -\sin x \, -\frac{x}{50}$$ $$f''(x) = -\cos x - \frac{1}{50}$$ and I want to find out how many roots it has, I have tried ...
### Does $\sum (-1)^n p_n$ converge to $0$ for even $n$?
Let $p_n$ be the $n$-th prime number (starting with $p_0=0 , p_1=2, ...$). I computed the zeros of the polynomial $$f_n(x) = p_nx^n + p_{n-1}x^{n-1} + ... + p_1x + p_0$$ for different $n$. This ...