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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.

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448 views

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 ...
11
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0answers
227 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely $$\dfrac{2\pi(x-\...
9
votes
0answers
177 views

Let $x_n$ be the (unique) root of $\Delta f_n(x)=0$. Then $\Delta x_n\to 1$

Note that by Cesaro's Theorem, we have as a consequence $$\frac{x_n}n\to 1$$ Consider $$r_n(x)=e^{-x}-\sum_{k=0}^n (-1)^k\frac{x^k}{k!}$$ and $$f_n(x)=(-1)^{n+1}e^{-x}r_n(x)$$ One can argue by $r_n(...
7
votes
0answers
49 views

Find the number of zeros of $f(z)=e^{z-1}-az$ inside unit disk, assuming $\mid a \mid >1$

This is an application of Rouche's theorem, I want to make sure I am doing it correctly: Let $f(z)=e^{z-1}-az$, where $\mid a \mid>1$ and $g(z)=-az$ Now, on the unit circle we have: $$\mid g(z) \...
7
votes
0answers
194 views

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 ...
7
votes
0answers
221 views

Roots of combination of trigonometric functions

Consider the following function of $\mathbb{R^+}^2$: $$f(s_1,s_2)=r_1^2\sin\big(\tfrac{1}{2}(s_2-s_1)\omega_1\big)\sin\big(\tfrac{1}{2}(s_2+s_1)\omega_2\big) + r_2^2\sin\big(\tfrac{1}{2}(s_2+s_1)\...
7
votes
0answers
127 views

Can we express the roots of all polynomials in terms of roots of some special polynomials?

We can describe the roots of quadratic equations in terms of addition, subtraction, multiplication, division, and the square-root function $\sqrt a$ which computes a root of the special polynomial $x^...
6
votes
0answers
157 views

Does $N_0(T) =N(T)$ where $N$ is the exact Riemann $\zeta$ zero counting function and $N_0$ is the approximate zero counting function imply the RH?

From A theory for the zeros of Riemann ζ and other L-functions The main new result is that the zeros on the critical line are in one-to- one correspondence with the zeros of the cosine ...
6
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0answers
130 views

Resolution of symbolic equations

It is well-known that CAS are able to perform operations like formal differentiation (relatively easily) and formal integration (via the Risch algorithm) in an algorithmic way. But is there anything ...
6
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0answers
87 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, z\in\...
6
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0answers
210 views

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 $n$. ...
5
votes
0answers
258 views

Find the roots of $x^3 -6x^2 +13x -12$

I am trying to find the roots of $$\tag{1} x^3 -6x^2 +13x -12$$ by applying the method outlined here (I think this Cardan’s method). Letting $y= x-2$, we can transform $(1)$ into $$\tag{2} y^3 + y -2=...
5
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0answers
147 views

Proof verification - A polynomial $P(x)$ has only real roots $\implies$ $P'(x)$ also has only real roots

Here's a problem that I've solved but I'm not very confident on my solution. Please check it there's any gap in my arguments. Also, is there a way to come up with a shorter proof ? Thank you. The ...
5
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0answers
179 views

Equations with Chebyshev polynomials

For a natural number n, let $r(x)$ be the polynomial $$r(x)=\prod_{k=1}^n(x-2\sin(\frac{\pi k}{2n+1})).$$ Then $-xr(2x)r(-2x)$ is Chebyshev polynomial of the first kind with integer coefficients. ...
5
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0answers
182 views

Irreducible polynomial such that its roots satisfy a given relation $x_3 = f(x_1,x_2)$

My general question is: What are the polynomials $f \in \Bbb Q[X,Y]$ such that there exists an irreducible polynomial $P \in \Bbb Q[X]$ having three distinct zeros $x_1,x_2,x_3$ that satisfy $x_3 = ...
5
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0answers
499 views

The probability that a random (real) cubic has three real roots

We can formalize the notion of the probability that a randomly selected quadratic real polynomial has real roots as follows: Suppose $R > 0$, and suppose the random variables $a, b, c$ are (...
5
votes
0answers
65 views

“gapped” polynomial leads to ring-shaped roots

Given a polynomial $$P(z)=\sum_{n=0}^N a_n z^n$$ with real coefficients distributed as a gaussian curve $a_n=\frac{1}{\sigma\sqrt{2\pi}}e^{(n-b)^2/2\sigma}$ ($b>0$). The sum of all the polynomial ...
5
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0answers
133 views

Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
5
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0answers
147 views

Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry $x,y&...
5
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0answers
120 views

Roots of a polynomial

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then $p_n(a)\...
5
votes
0answers
126 views

Can even degree Legendre polynomials have roots in common?

I'm wondering whether the Legendre polynomials $P_m(x)$ and $P_{m+2k}(x)$, with $m$ even and $k \in \mathbb{N}^+$, can have common roots. For $k=1$ it is straightforward to show that there are no ...
5
votes
0answers
648 views

Prove equation has only one root in a specific interval

Prove that the following equation has only one solution in the interval $[-\text{min}(a_i), +\infty]$: $f(x) = \left(\sum_{i=1}^n \frac{1}{a_i + x}\right)\times \left(\sum_{i=1}^n \frac{a_i b_i}{(a_i ...
5
votes
0answers
94 views

How does the polynomial transformation $P(x) \mapsto P(x) + c$ alter the roots of that polynomial? Specifics inside.

Consider a real quadratic polynomial $Q_k(x) = (x-\nu)(x-\omega_k) - g_k^2$. I can interpret $Q_k(x)$ as a translation of the polynomial $$ (x-\nu)(x-\omega_k) = \left(x-\frac{\nu+\omega_k}{2}\right)^...
4
votes
0answers
90 views

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) ...
4
votes
0answers
41 views

Is there any numerical data or theorems on the minimum distance between consecutive zeros of the zeta function?

Title basically. I'm looking for any results, papers, or data pertaining to the distribution of zeros of the zeta function on the critical line. Some numerical data which has a max height $T$ and $\...
4
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0answers
61 views

Showing two splitting fields are different

I am trying to do problem 3.28 from the Algebra questions on this site. It says the following: How would you find the Galois group of $x^3+2x+1$? Adjoin a root to $\mathbb Q$. Can you say something ...
4
votes
0answers
93 views

Can we write $\ln(x) $ as an infinite sum of $n$ th roots?

Is there a real sequence $0 \leq a_n $ such that for $x > 1 $ we have : $$\ln(x) = a_0 + \sum_{n=1}^{\infty} (-1)^n \cdot a_n \cdot x^{1/n}$$ Or $$\ln(x) = a_0 + \sum_{n=1}^{\infty} (-1)^{1+n} ...
4
votes
0answers
105 views

A rational function with hidden symmetry and alternating poles and zeros.

Upon answering a question about an equivalence of two binomial sums I have noted that a naturally appearing function has some interesting properties. Consider the function: $$ f(m,n_1,n_2;z)=\frac{1}{...
4
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0answers
61 views

Looking for references about a graphical representation of the set of roots of polynomials depending on a parameter

Answering, some time ago, to this question : Change in eigenvalues on changing one entry of a matrix, I had the idea of a graphical representation of roots of polynomial equations $P(x,a)=P_{a}(x) \in ...
4
votes
0answers
116 views

Understanding string diagram page 115 Humphreys Lie algebras

Trying to understand the figure on page 115 of Humphreys 'Introduction to Lie Algebras and Representation Theory'. He takes $V(\lambda)$ as a representation of $\mathfrak{sl}_3$. Where $\lambda = 4\...
4
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0answers
94 views

The zeros of $\frac{\sin(\sqrt{z})}{\sqrt{z}}$

It is known that the function $f(z) = \frac{\sin(\sqrt{z})}{\sqrt{z}}$ is entire. What are it's zeros? I think that $\sqrt{z} = k \pi$ with $k \in \mathbb{Z}\setminus\{0\}$. Then: For instance $ \...
4
votes
0answers
51 views

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 ...
4
votes
0answers
90 views

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 ...
4
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0answers
2k views

The equations $x^3+5x^2+px+q=0$and $x^3+7x^2+px+r=0$ have 2 common roots, then find the third root of both equations

The equations $x^3+5x^2+px+q=0$and $x^3+7x^2+px+r=0$ have 2 common roots, then find the third root of both equations From the first equation we can say, $\alpha\beta+\beta\gamma+\gamma\alpha=p/...
4
votes
0answers
137 views

Is ∛ symbol for principal third root or real third root?

I thought I knew root notation quite well, but now I am confused, and various sources say different things. So how much is $\sqrt[3]{-27}$? Wikipedia and google both clearly state that the solution ...
4
votes
0answers
92 views

How did the Babylonians derive the Secant Method without algebra?

So, I was just reading about the Secant Method and came across the sentence that it: "developed independently of Newton's method, and predates it by over 3,000 years.". This would mean that it was ...
4
votes
0answers
188 views

Correcting Gauss's proof of FTA. Need verification

In his doctoral thesis, Gauss gave a proof of fundamental theorem of algebra for real polynomials, based on geometric arguents. Later in his life he expanded the proof to complex polynomials. A nice ...
4
votes
0answers
170 views

Find the complex (or real) roots of $e^{\frac{3 x}{2}}+2 \cos \left(\frac{\sqrt{3} x}{2}\right)$

Define for natural $n\geq 2$ $$G(x,n)= \sum _{k=0}^\infty \frac{x^{k n}}{(k n)!}= \frac{\sum _{k=0}^{n-1} e^{x e^{\frac{2 i \pi k}{n}}}}{n}= G(x e^{\frac{2 i \pi}{n}},n)= \prod_{m=1}^\infty \left(1+\...
4
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0answers
85 views

On the location of the roots of a polynomial

Consider the following two polynomials \begin{align} p(s)&:=s^n+\alpha_{n-1}s^{n-1}+\cdots+\alpha_1s+\alpha_0,\\ q(s)&:=s^{n-1}+\alpha_{n-1}s^{n-2}+\cdots+\alpha_2 s+\alpha_1, \end{align} ...
4
votes
0answers
98 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ e_q(z)=\prod_{k=0}^...
4
votes
0answers
169 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ A_{...
4
votes
0answers
165 views

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 ...
4
votes
0answers
1k views

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 ...
4
votes
0answers
841 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - (...
3
votes
0answers
38 views

Solving $\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du=0$ (an extension to the Ramanujan-Soldner constant)

For $u,x>0$, let $P$ be the function given by $$P(x)=\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du\tag1.$$ Is there a closed form for the positive root of $P(x)$, denoted by $\nu$? Can it be ...
3
votes
0answers
65 views

$\cos\frac\pi{n}$ Analytic expression

I recently found out that $$\sin\frac\pi5=\frac12\sqrt{\frac{5-\sqrt5}2}$$ Which means that $$\cos\frac\pi5=\frac{1+\sqrt5}4$$ I also recently found that if $n\in\Bbb N$, $$\sin nx=\sin x\,U_{n-1}(\...
3
votes
0answers
57 views

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 ...
3
votes
0answers
68 views

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 ...
3
votes
0answers
71 views

Finding how many solutions a function has

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 ...
3
votes
0answers
113 views

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 ...