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 (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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11
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1answer
2k views

Derivation of asymptotic solution of $\tan(x) = x$.

An equation that seems to come up everywhere is the transcendental $\tan(x) = x$. Normally when it comes up you content yourself with a numerical solution usually using Newton's method. However, ...
5
votes
2answers
403 views

Roots of the incomplete gamma function

Is there any way that one can describe all the roots of the incomplete gamma function $\Gamma(n,z)$, for $n\in \mathbb{N}$, analytically?
2
votes
0answers
81 views

When does $f(x)/f'(x)$ have a first-order root?

Actually let $g(x)=0$ when $f(x)=0$ otherwise $g(x)=f(x)/f'(x)$. Seems clear to me that if $x_0$ is an $n$-order root of $f(x)$ where $n$ is a positive integer, and $f(x)$ can be expressed as a ...
4
votes
0answers
362 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)\\ & {} - ...
0
votes
1answer
427 views

Spherical Bessel Zero's

I was wondering if there is a known closed form solution for the zeros of the Spherical Bessel Functions. While doing a Quantum assignment I came across them as a solution for the spherical infinite ...
4
votes
2answers
267 views

roots of complex polynomial - tricks

What tricks are there for calculating the roots of complex polynomials like $$p(t) = (t+1)^6 - (t-1)^6$$ $t = 1$ is not a root. Therefore we can divide by $(t-1)^6$. We then get $$\left( ...
10
votes
4answers
340 views

sum of reciprocals of derivative of polynomial at its roots

If $P(x)$ is a polynomial of degree $n > 1$ with only simple roots $a_1,\ldots,a_n$, is it true that $\frac 1{P'(a_1)} + \cdots + \frac 1{P'(a_n)} = 0$, and, if so, what is the proof? I ...
0
votes
1answer
176 views

Determining the density of roots to an infinite polynomial

Consider a polynomial defined by its roots: \begin{equation} P(z; \mathbf{S}) = \Pi_{\theta_j \in \mathbf{S}} (z - \exp({2 \pi i \theta_j}) ) \end{equation} where $\mathbf{S}$ is a set of numbers. ...
11
votes
1answer
2k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
1
vote
1answer
198 views

root of sum of exponentials

I am curious to know which values of $t \gt 0$ solve the following equation in terms of the constants $a,b,c$. $a e^{-2 b t} - e^{-2 t} + c e^{-3 b t} + c e ^{- 3 t} = 0$ where $a \gt 1, b \gt 1, c ...
0
votes
1answer
296 views

Clarification when using the Bisection method

I understand how the Bisection method works: you take an interval and test the end-points and the mid-point. Somewhere in those intervals, there will be a root. You keep swapping the mid-points and ...
1
vote
2answers
112 views

Converting polynomials to depressed form

Given a polynomial of any degree $\sum_{i=0}^n a_ix^i$ can it be proven that the substitution $x=t-$${a_{n-1}}\over{na_n}$ will convert the equation to depressed form $b_nt^n$ + $\sum_{i=0}^{n-2} ...
3
votes
1answer
65 views

Number of roots $|f(|x|)|$ has according to $f$

If $f$ has one root on $(-\infty,0)$ and two distinct roots on $(0,+\infty)$ and $f(0)=-1$, how many roots does $|f(|x|)|$ have? I know graph of $|f(|x|)|$ should be in quadrant I because $x$ ...
9
votes
1answer
632 views

If a rational function is real on the unit circle, what does that say about its roots and poles?

While doing a bit of self study, I ran across a situation whose wording confused me. Suppose $R(z)$ is some rational function which is real on the circle $|z|=1$ in the complex plane. The question ...
3
votes
2answers
611 views

roots of minimal and characteristic polynomial

Why is it, that for the matrix $A \in \text{Mat}(n\times n, \mathbb{C})$ the characteristic polynomial $\chi_A(t)$ and the minimal polynomial $\mu_A(t)$ have same roots? Since $\chi_A(t) = \mu_A(t) ...
5
votes
2answers
156 views

How many positive roots does the equation $a^x=x^a$ have?

Let $a\in (1,e)\cup(e,\infty).$ I'd like to show that the equation $a^x=x^a$ has exactly two positive solutions, and one is larger and one smaller than $e.$ Is it even possible to show? I think I've ...
5
votes
4answers
545 views

find the least a, for which two equations have a common root

Could you help me out please. I have two equations: $2x^2-3x+1=0 $ and $ 2x^2-(a+3)x+3a=0$ I need to find the least $a$ for which these two equations have a common root. At a first glance I thought ...
4
votes
3answers
251 views

If $f(x)$ and $f(x)-x$ have only one real root, then $f(f(x))-x$ has only one real root.

First edition was: Let $f(x)$ be a polynomial such that $f(x)$ and $f(x)-x$ have only one real root. How to prove, without derivatives, that $f(f(x))-x$ also has only one real root? Second edition: ...
1
vote
2answers
81 views

Normalization of a univariate polynomial with real algebraic coefficients

Consider a polynomial in one variable $x$ with irrational coefficients which are algebraic, i.e., they have a defining polynomial. As an example, take $p(x) = (x-3)(x-\sqrt{2}) = ...
2
votes
1answer
398 views

Bound the complex roots of a polynomial above

We consider $P(z)=a_{0}+a_{1}z+\cdot+a_{n-1}z^{n-1}+a_{n}z^n$, with $a_{0},\ldots,a_{n-1},a_{n} \in \mathbb{C}$ and $a_{n}\neq0$. Let $R=\max_{0\leq k\leq n-1}\left | \frac{a_k}{a_n} \right |$ and ...
3
votes
3answers
502 views

Algebraic conjugates

Suppose $L/K$ is an algebraic field extension. Take $\alpha_1 \in L$. Then $\alpha_1$ has minimal polynomial $f(x)$ over $K$. Let $\alpha_2, ... \alpha_k$ be the other roots of $f$ in $L$. The ...
2
votes
2answers
300 views

Proof existence of field extension of $\mathbb{F}_p$ containing the $r$-th primitive root of unity

I have to show the following: Let $p$ be a prime and $r \in \mathbb{N}$ with $\gcd(r,p)=1$. Prove the existence of a field extension $E$ of $\mathbb{F}_p$ which contains an $r$-th primitive root ...
4
votes
1answer
150 views

True or False: if $n$ is not even then $P(x)=x^n+ax^2+b$ has at most 3 roots

I have a homework question which is: True or False: if $n$ is not even then $P(x)=x^n+ax^2+b$ has at most 3 roots I know that the version of $n$ being even is true via some recursion and ...
2
votes
0answers
181 views

Solution to polynomial equations with non-radicals

For degree 1, 2, 3 and 4 there is an "extended a,b,c-formula" (like the one we learn in middle or high school, http://en.wikipedia.org/wiki/Quadratic_equation) for the solution to a polynomial ...
6
votes
1answer
305 views

Why are primitive roots of unity the only solution to these equations?

I was led by this question to the following problem: Find $n$ complex numbers $\lambda_1\dots\lambda_n\in\mathbb{C}$ that satisfy $$\begin{align} \sum_i\lambda_i & =0\\ \sum_i\lambda_i^2 ...
2
votes
3answers
953 views

Newton's method and trig functions on a computer

I'm trying to use Newton's method to find roots for the function $A \cos(\Theta_2 - \Theta_1) + B \sin(\Theta_1)$. (That is, iterate $x_{i+1} = x_i - f(x_i) / f'(x_i)$). I've got a working ...
5
votes
1answer
967 views

Using the fifth roots of unity to find the roots of $(z+1)^5=(z-1)^5$

The question I am working on starts of with: Find the five fifth roots of unity and hence solve the following problems I have done that and solved several questions using this, however ...
2
votes
6answers
562 views

What are the common solutions of $x^2+y=31$ and $y^2+x=41$?

A friend asked me if I have a certain algorithm to solve $x^2+y = 31$ and $y^2+x=41$ simultanously. We found the solutions but we didn't find a way to solve both equations. Any ideas?
0
votes
1answer
91 views

Guessing bounds for the roots of a function, and counting roots within those bounds

In my book, it's written that we can guess how many roots an equation might have and where they approximately are by its graph or the table of function values without trying to solve it. There's an ...
1
vote
1answer
118 views

Algorithm to approximate roots using blackbox root counting function

I am looking for an algorithm to do the following. Given: An interval $[a,b]$. A black box function $n(x)$ which returns the number of roots (zeros) of a function that lie to the left of $x$. That ...
0
votes
1answer
427 views

Point projection on curve

Point projection on Bézier curves can be easily accomplished using Newton Iteration to try to minimize the dot product between the vector connecting the point P and its projection on curve C and the ...
1
vote
2answers
112 views

Efficient computation of the trajectory of roots of a parameterized polynomial

Let $N(s)$ and $D(s)$ be two polynomials in $s \in \mathbb C$ of degrees $m$ and $n$, respectively, with $m<n$. Consider the polynomial equation $$P(s) = N(s) + kD(s) = 0,$$ where $k > 0$. For ...
38
votes
4answers
935 views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
0
votes
1answer
73 views

First derivative

What would be the further steps for the case like this: I am finding the first derivative of a function: $f(x) = \ln(1+x^2)$ So the procedure would then be: $f'(x) = \frac{2x}{1+x^2}$. $f'(x) ...
6
votes
3answers
646 views

What we can say about $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$?

Problem: How we can strictly prove $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ can't be 2? Can $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ have the value expressed by complex numbers? (See below, in ...
0
votes
1answer
240 views

Sum of the polynomial roots raised to a power. How to prove?

Problem: If we have a polynomial $f$ with a derivative $f\,'$ and quotient $q$ function defined as: $$q(x)=\sum_{i=1}^{\infty}a_ix^{-i}=\frac{f\,'(x)}{f(x)},$$ and the roots of $f$ are ...
0
votes
1answer
4k views

Relation betwen coefficients and roots of a polynomial [duplicate]

Possible Duplicate: Create polynomial coefficients from its roots I am reading the first chapter titled Numerical Solutions Of Equations And Interpolation by K.A. Stroud (Advanced ...
3
votes
5answers
364 views

How to compute the characteristic polynomial of $A$

The matrix associated with $f$ is: $$ \left(\begin{array}{rrr} 3 & -1 & -1 \\ -1 & 3 & -1 \\ -1 & -1 & 3 \end{array}\right) . $$ First, I am going to find ...
9
votes
1answer
240 views

Roots of a complex polynomial

Let $f(z)=a_0+\cdots+a_n z^n$ be a complex polynomial with $a_j = \alpha_j + i \beta_j$ (with $\alpha_j, \beta_j \in \mathbb{R}$). Let $P(z)=\alpha_0 + \cdots + \alpha_n z^n$ and $Q(z)=\beta_0 + ...
48
votes
4answers
3k views

Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?

Problem: Find $x$ in $$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$ Trick: $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so, $x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and, ...
2
votes
1answer
143 views

polynomials that have distinct roots

I have known some Sufficient Condition for All the Roots of a Polynomial To Be Real. Is there any sufficient condition that a polynomial of degree $n$ has $n$ distinct roots? For $n=2$, it is trivial. ...
0
votes
1answer
171 views

Negative real parts and of the solution of a polynomial and stable matrices

Can someone give me an idea how to show that the real part of the solutions of the equation $$ x^3+bx^2+xc+d$$ arenegative iff $b>0,d>0$ and $-bc<-d$ ? This question is related to proving ...
8
votes
3answers
2k views

Root or zero…which to use when?

This may seem like a very basic question, but: What exactly is the difference between a root of a polynomial, and a zero? Of course I realise that they are technically exactly the same thing, but ...
3
votes
1answer
182 views

How to find the roots of $f(x)= \ln( \frac{x+1 }{x-2})$?

I can't solve this equation: $$\ln\left(\frac{x+1}{x-2}\right) = 0.$$ I do: $$\begin{align*} \ln \left( \frac{x+1}{x-2} \right)&=0\\ \frac{x+1}{x-2} &= 1 \\ x+1&=x-2 \\ ...
0
votes
1answer
143 views

Roots of elementary monomials

Let $m_\lambda(X_1,X_2,...X_N)$ be a monomial symmetric function with partition $\lambda$. For example: $$ m_{(3,1,1)}(X_1,X_2,X_3) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3 $$ Is there a general ...
2
votes
1answer
1k views

How many roots of a polynomial have positive real part?

I am given an exercise with three polynomials, and we have to find the number of roots of the first one that lie in the unit disk, the number of roots that lie in some region, e.g. those that lie in ...
6
votes
1answer
222 views

Complex Logs and Roots of Unity

I need to find all the solutions to the following using logarithms: $(e^z-1)^3=1$ where z is a complex number. I am told that using roots of unity I can break this equation down but I must be missing ...
1
vote
3answers
146 views

Adjunction of a root to a UFD

Let $R$ be a unique factorization domain which is a finitely generated $\Bbbk$-algebra for an algebraically closed field $\Bbbk$. For $x\in R\setminus\{0\}$, let $y$ be an $n$-th root of $x$. My ...
0
votes
1answer
114 views

How do I find p for equations of the form $\sum \limits_i \frac{a_i}{b_i^p} = 1$

The problem I'm facing is solving the following equation for $p$ given the constants $a_i$ and $b_i$: $$ \sum_i \frac{a_i}{b_i^p} = 1 $$ Is there a general technique that would allow me to find a ...
4
votes
1answer
107 views

Roots of $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ in the interval $[0,1]$

Does the polynomial $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ have exactly one root in the interval $[0,1]$ for all non-negative integers $n$? It has at least one root because $f_n(0) = -1$ and ...