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.
4
votes
3answers
338 views
Square roots — positive and negative
It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive ...
40
votes
4answers
2k 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,
...
40
votes
5answers
823 views
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
12
votes
3answers
1k views
Why is it so hard to find the roots of polynomial equations?
The question that follows was inspired by this question:
When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic ...
8
votes
1answer
792 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, ...
4
votes
1answer
628 views
Why not write the solutions of a cubic this way?
For the solution of the cubic equation $x^3 + px + q = 0$ Cardano wrote it as:
$$\sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + ...
23
votes
3answers
626 views
Galois groups of polynomials and explicit equations for the roots
Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
7
votes
4answers
279 views
Algorithms for “solving” $\sqrt{2}$
The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
29
votes
5answers
1k views
Continuity of the roots of a polynomial in terms of its coefficients
It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
4
votes
6answers
619 views
Fastest Square Root Algorithm
What is the fastest algorithm for finding the square root of a number?
I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
1
vote
1answer
227 views
Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$
The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has
(A) exactly two distinct real roots
(B) exactly three distinct real roots
(C) exactly four distinct real roots
(D) six distinct real roots
14
votes
5answers
3k views
Roots of Legendre Polynomial
I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
Are the roots always simple (i.e., multiplicity $1$)?
...
4
votes
6answers
142 views
Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$
Where do I start to solve a equation for x like the one below?
$$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$
After squaring it, it's too complicated; but there's nothing to factor or to ...
3
votes
5answers
130 views
How to find the number of real roots of the given equation?
The number of real roots of the equation $$2 \cos \left( \frac{x^2+x}{6} \right)=2^x+2^{-x}$$ is
(A) $0$, (B) $1$, (C) $2$, (D) in finitely many.
Trial: $$\begin{align} 2 \cos \left( ...
6
votes
1answer
103 views
Location of zeros of an analytic function
Describe the approximate locations of the zeros of the function
$$
f(z) = e^{iz}+e^{-iz}+e^z
$$
lying outside the circle $|z|=R >>1$.
Another prelim problem. For Rouche's theorem we need to ...
2
votes
2answers
252 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?
3
votes
1answer
227 views
How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?
A few days ago, I had a vague question in my mind about "matrix methods" for finding roots of a polynomial. Now I can ask at least a semi-precise question, thanks to the post
How to calculate complex ...
29
votes
4answers
778 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 ...
9
votes
2answers
299 views
Zeros of Fourier transform of a function in $C[-1,1]$
I am trying to prove the following:
Let $g \in C[-1,1]$. Then the function $$G(z) = \int_{-1}^1 e^{itz}g(t)dt$$ has infinitely many zeros.
I know that $G(z)$ is entire and $\lim_{x \to \pm ...
6
votes
2answers
169 views
Is $\sqrt 7$ the sum of roots of unity?
Let $a_n$ and $b_n$ be 2 sequences of $n$ rationals.
Is it possible that $\sqrt 7 = \sum_{m=1}^{n} a_m (-1)^{b_m}$ ? Is it possible that $\sqrt{17}$ = $\sum_{m=1}^{n} a_m (-1)^{b_m}$ ?
How to ...
6
votes
3answers
482 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 ...
15
votes
3answers
1k views
On applying the quadratic formula to a first-degree equation
You're probably thinking, "Why?" Please let me explain...
It is (very) well-known that
$$ \forall (a,b,c,x) \in \mathbb{C}^* \times \mathbb{C}^3: ax^2 + bx + c = 0 \Leftrightarrow x = \frac{-b \pm ...
11
votes
2answers
432 views
Number of roots of $x^a-1=0$ with $a \in \mathbb{C}$
It is well known that $x^2-1=0$ has two roots in $\mathbb{C}$, namely $\pm 1$. In general $x^n-1=0$ has exactly $n$ roots in $\mathbb{C}$. But what happens when $n$ is non integer (rational or real or ...
4
votes
4answers
769 views
Finding all complex zeros of a high-degree polynomial
Given a large univariate polynomial, say of degree 200 or more, is there a procedural way of finding all the complex roots? By "roots", I mean complex decimal approximations to the roots, though the ...
10
votes
6answers
466 views
Complex roots of $z^3 + \bar{z} = 0$
I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre.
Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
5
votes
1answer
262 views
Finding all roots of polynomial system (numerically)
I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton ...
9
votes
1answer
958 views
Using Vieta's theorem for cubic equations to derive the cubic discriminant
Background:
Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then
$$\begin{eqnarray*}
-p &=& x_1 + x_2 ...
9
votes
2answers
414 views
Complex Zeros of $z^2e^z-z$
Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? ...
6
votes
1answer
236 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
$$\begin{align}
\sum_i\lambda_i & =0\\
\sum_i\lambda_i^2 ...
4
votes
3answers
260 views
Is there an algorithm to find the roots of high-order polynomials?
It is not generally possible to determine the roots of a polynomial whose grade is bigger than 4 in terms of roots and basic operations. But I heard, that it is possible to give a criteria whether a ...
3
votes
2answers
74 views
Convergence of Roots for an analytic function
Show that the roots of
$$
f(z) = z^n+z^3+z+2 =0
$$
converge to the circle $|z|=1$ as $n \to \infty$.
3
votes
3answers
450 views
How to tell if a quartic equation has a multiple root.
Is there any way to tell whether a quartic equation has double or triple root(s)?
$$x^4 + a x^3 + b x^2 + c x + d = 0$$
3
votes
3answers
149 views
Using intermediate value arguments at limits rather than finding explicit bounds
Again, I apologize for what looks like a very narrow question. But there's possibly a general principal at work here that I'm not grasping.
I understand the answer provided for exercise 3 in chapter ...
0
votes
0answers
46 views
Upperbound on the number of Isolated zeros of a bivariate polynomial
Let $F(x,y)$ be a bivariate polynomial, of degree n. Hence:
$F(x,y) = \underset{i+j \leq n}{\sum_{i=0}\sum_{j=0}}a_{ij}x^{i}y^{j}$
Can there exist an upperbound for the number of isolated zeros for ...
8
votes
1answer
330 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
56 views
Finding root using Hensel's Lemma
Hensel's Lemma calculates root of a polynomial $\in \mathbb{Z}_p[X]$ but is there any other significance to other branches of mathematics or outside mathematics? Why is finding root of ...
3
votes
1answer
96 views
Understanding accuracy of Newton's Method
In a numerical analysis book I'm reading it says that using the Newton error formula we can find an expression for the number of correct digits in an approximation using Newton's Method.
Here's the ...
3
votes
2answers
604 views
Fast and robust root of a cubic polynomial with constraints
I'm looking for a fast and robust method for finding a root of a cubic polynomial
$x^3 + px^2 + qx + r$
To make the search more robust and faster, I'd like to leverage these properties:
The ...
3
votes
2answers
197 views
Quaternions as roots
So, I StumbledUpon this really cool site and the last picture looked almost as if it had 3D structure. This reminded me of another website where I saw pictures of the order-8 Mandelbulb. I got to ...
2
votes
2answers
124 views
looking for a technique to solve an indefinite integral of one over the square root of a cubic polynomial
I am looking for a technique to solve an indefinite integral of
$$
\int \frac{dx}{\sqrt{ax^3+bx^2+cx+f}}
$$
I honestly have no idea where to start with this and I cannot find anything like this in ...
2
votes
1answer
764 views
How many roots of a polynomial have positive real part?
I am given an exercise with three polynomials, and we have to find the number of roots of the first one that lie in the unit disk, the number of roots that lie in some region, e.g. those that lie in ...
1
vote
2answers
96 views
Solve a quadratic matrix equation?
Given a known symmetric matrix $M$, vector $\vec{v}$ and scalars $a$ and $b$, I'm trying to solve for a scalar $x$ such that:
$\vec{v}^T(M+(ax+b)I)^{-1}\vec{v} - ...
1
vote
3answers
123 views
Proving square root of a square is the same as absolute value
Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't ...
1
vote
3answers
149 views
Equations - Solving for x
I have this problem:
$$9x^3 - 18x^2 - 4x + 8 = 0$$
However, I'm not sure how to find the values of $x$. I moved the 8 over and factor out an $x$, but the trinomial it created can't be factored. ...
1
vote
2answers
97 views
Efficient computation of the trajectory of roots of a parameterized polynomial
Let $N(s)$ and $D(s)$ be two polynomials in $s \in \mathbb C$ of degrees $m$ and $n$, respectively, with $m<n$. Consider the polynomial equation $$P(s) = N(s) + kD(s) = 0,$$
where $k > 0$.
For ...
1
vote
1answer
95 views
Proving that zeros of a function are first-order
Let $c_{L},c_{T},\omega$ be positive constants with $c_{L}>c_{T}$. Define
$$p=\sqrt{\frac{\omega^{2}}{c_{L}^{2}}-\xi^{2}}\qquad q=\sqrt{\frac{\omega^{2}}{c_{T}^{2}}-\xi^{2}}$$
Consider the ...
0
votes
4answers
86 views
For $\sqrt[3]{-1+i}$, is $r$ (when put in polar form) $\sqrt[6]{2}$?
And when you put that into the nth root form... It becomes $2^{1/18}\cos\theta + 2^{1/18}\sin\theta$?
$n$th root form given is: $\sqrt[n]r\cdot\cos(\theta+2\pi k)n$
0
votes
4answers
121 views
How do I form this equation?
If $A$ and $B$ are the root of the equation $3x^2-4x-9=0$, what is the equation whose roots are
$(A+3)/(A-3)$ and $(B+3)/(B-3)$
0
votes
1answer
127 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. ...
-9
votes
2answers
163 views
What is the Positive Root of the Equation? [closed]
Given that: $f(x)=-x^2 - 5x + 2$ and $g(x) = f(x-)$ what is the Positive Root of the equation $f(x) = g(2)$?
This is the Question that my math teacher put on the board, she said if we got the answer ...
