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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2
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1answer
209 views

Formal Derivative and Multiple roots

I am currently really stuck on the following problem: Prove that if f(x) in Fp[x] and Df = 0 (where D : Fp[x] → Fp[x] is the formal derivative) then there exists g(x) in Fp[x] such that f(x) = g(x)^p ...
4
votes
2answers
84 views

Question about roots

Let $a,b,c$ be roots of equation $x^3-6x^2+kx+k=0$,and $(a-1)^3+(b-2)^3+(c-3)^3=0$. how to compute $a,b,c,k=?$ if we do work equivalently as to find out the solution: ...
6
votes
2answers
440 views

Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
3
votes
8answers
4k 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 ...
0
votes
1answer
66 views

Consider polynomial $q(x,y)=(2x+3y)^2-1$, how to show that it has roots with arbitrary values of x,y?

Given the following polynomial $q(x,y)=(2x+3y)^2-1$. How would I show that it has roots with any large $x,y$?
1
vote
1answer
37 views

Maximal Distinct Roots in $F_q$

Let $a\in F_q[x]$, and let $r(\cdot)$ denote the number of distinct roots over $F_q$. For any $i|q$, prove that $$ \max_{\deg(a)=1}r(x^i-a)=r(x^i-x) $$
5
votes
2answers
2k views

Find the roots of a polynomial using its companion matrix

I would like to find the roots of a polynomial using its companion matrix. The polynomial is ${p(x) = x^4-10x^2+9}$ The companion matrix $M$ is $M={\left[ \begin{array}{cccc} 0 & 0 & 0 ...
8
votes
2answers
410 views

closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
17
votes
1answer
253 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but ...
4
votes
1answer
134 views

Mean values theorem and countable sets

The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u)$$ Assume that $f$ is a ...
1
vote
1answer
53 views

Root of sum of shifted polynomials

For an arbitrary positive odd integer $k$, I would like to obtain an expression for the root $x_{root} \in \mathbb{R}$ of the following polynomial $$p(x) = \sum_{i=1}^N (x-x_i)^k,$$ where $x_i\in ...
10
votes
1answer
338 views

How to solve $x!=5^x$?

Or, more generally, $$\Gamma (x+1)=\int_0^{\infty}t^{x}e^{-t}dt=p^x$$ with $p \in \mathbb{Z}^+$ and $x \in \mathbb{C}$. Perhaps begin with $\large p^x=p^x \lim_{n \rightarrow ...
1
vote
1answer
27 views

How to get started on showing the conditions that $ax+by+cz=0$

I am looking at this question from Hardy's book, A Course of Pure Mathematics and have no idea where to begin. I was wondering, what is the first step to deriving the conditions? Question What are ...
1
vote
0answers
133 views

Given relations of coefficients and $m$ zeros of a complex polynomial, find the polynomial of degree $2n$ and $m \geq n$.

Given relations of coefficients and $m$ zeros of a complex polynomial (coefficients are complex), find the polynomial of degree $2n$ and $m \geq n$. For examples, we are finding $P(x)=C_{2n}x^{2n} + ...
0
votes
1answer
57 views

Is there an analytic solution to the following equation

I have the following general equation in $x$ $$a\cos(b - cx) - d\cos(e - fx) = 0$$ with constants $a,b,c,d,e,f$. Is there an algerbraic solution to this or only a numeric one?
1
vote
1answer
493 views

Root Locus, Meaning of the Roots?

I'm studying control theory and I encountered the root locus, I know that It plots the roots of the characteristic equation but i've some questions. What is the physical meaning of the Roots of the ...
0
votes
1answer
201 views

Change of argument of $\exp(z)-z$ on each side of a square

Show that as the positive integer $N$ tends to $\infty$, the change in argument of $e^z − z$ is bounded on $3$ sides of the square with corners $ \pm 2\pi N$ $\pm 2\pi iN$ but is unbounded on the ...
4
votes
3answers
164 views

Multiplying a square root by a non-square root

This is not something I do very often, so I'm a bit dicey on the rules. I just want to make sure that I understand things right... $$-\frac{1}{2}\cdot \sqrt{\frac{2}{5}} = -\sqrt{\frac{1}{4}}\cdot ...
1
vote
2answers
351 views

Proof $ \sqrt{1 + \sqrt[3]{2}} $ is irrational using the theorem about rational roots of a polynomial

I'm having trouble with this specific problem at the moment. The theorem states that if $n/m$ is a rational root of a polynomial with integer coefficients, the leading coefficient is divisible by m ...
4
votes
1answer
686 views

Argument principle: number of zeroes of $f(z)=\cos(z)-1 +z^2/2$ in the unit disk

I am trying to work on this old qual exam. Here is the question: Find the number of roots (counting multiplicities) of the function $$f(z)=\cos(z)-1 + \frac{z^2}{2}$$ inside the domain $\vert ...
0
votes
3answers
249 views

Common method of calculating zero places of quadratic and linear function.

Very basic stuff from school we know that we can calculate zero places of quadratic function which has form $ax^2 + bx + c$ and we assume that $a \neq 0$, now what if $a=0$? Why can't we use delta to ...
2
votes
1answer
99 views

About canonical factors for Weierstass infinite products.

I was reading the proof of below theorem (p.145 complex analysis Elias M.Stein): Given any sequence $\{a_n\}$ of complex numbers whit $|a_n| \to \infty $ as $n \to \infty $, there exists an entire ...
4
votes
1answer
220 views

Number of roots of a complex equation/ Rouche's theorem

For $n\geq2$ consider the equation $z^n+z+n=0$ for $z\in \mathbb C$. Show that if $k$ is an integer with $1\leq k \leq n$ then inside the sector $$ S_k=\left\{z\in \mathbb C: 0< Arg(z) < ...
1
vote
2answers
70 views

Exponential equation with three summands

I had a simple looking math problem the other day: Solve for $y(x) = 0$: $$ 10^{2x} - 101 \cdot 10^x + 100 = 0$$ Since I have three summands, I cannot just put them to either side of the ...
1
vote
1answer
51 views

Pick out the case(s) which ensure that the polynomial $p(\cdot)$ has a root in the interval $[0, 1]$

Please help me to solve the problem below. Consider the polynomial $p(x) = a_0 + a_1x + a_2x^2+\dots+ a_nx^n$, with real coefficients. Pick out the case(s) which ensure that the polynomial ...
1
vote
1answer
689 views

Complex Analysis - Argument Principle vs. Rouche's Theorem

The Argument Principle Suppose a function $f$ is meromorphic on an open set that contains a circle $C$ and its interior. Further assume that $f$ has no zeroes on $C$ (but may have zeroes in the ...
0
votes
2answers
127 views

Plot of a Bessel function if possible

i would like to know where i could find a plot of $$ J_{ia}(2\pi i)$$ (1) using Quantum mechanics i have conjectured that if $ a= \frac{x}{2} $ and $ i= \sqrt{-1} $ then $$ J_{it}(2\pi ...
6
votes
1answer
396 views

Complex Analysis - Location of roots of a polynomial

How many roots does the polynomial $z^4 + 3z^2 + z + 1$ have in the right-half complex plane (i.e. $Re(z) \gt 0$)? I honestly can't think of how to approach the problem as it seems different from the ...
3
votes
2answers
123 views

How to show that there exists a root of $f(x)=0$?

Let $f(x)=\sum_{k=0}^n a_k x^k$, where $a_k$'s satisfy $\sum_{k=0}^n \frac{a_k}{k+1}=0$.Show that there exists a root of $f(x)=0$ in the interval $(0,1)$.
3
votes
1answer
297 views

Root Finding Algorithm for Discrete Functions

I was recently working with functions of the form $$N - \sqrt{\frac{N}{x}}\cdot\left\lfloor \frac{N}{\sqrt{N/x}}\right\rfloor + \sqrt{\frac{N}{x}} - \left\lfloor \sqrt{\frac{N}{x}}\right\rfloor$$ ...
3
votes
0answers
66 views

Zeros of $ \frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$

What would be the zeros of the following function? $$ \frac{1}{B(xi)^{1/2}}((iA)^{ix})(ix)^{ix}+ \frac{1}{B(-xi)^{1/2}}((-iA)^{-ix})(-ix)^{-ix}=H(x)$$ This function is real and I believe it is equal ...
1
vote
1answer
257 views

Multiple roots of a polynomial in two variables

Let $F\in\mathbb{C}[X,Y]$ be an irreducible polynomial and $n\in \mathbb{N}$, $n\ge1$, $p_i\in\mathbb{C}[X]$ for $0\le i\le n$, such that $$F(X,Y)=\sum\limits_{i=0}^{n}p_i(X)Y^{n-i}.$$ Let ...
18
votes
4answers
754 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
2
votes
0answers
250 views

Using the duplex method to calculate square roots

I have been assigned to find out how a calculator figures out square roots, so far the shortest thing I can see is "the duplex method". But the thing is that the explanation on Wikipedia makes no ...
9
votes
2answers
511 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 ...
0
votes
0answers
254 views

Computing square roots modulo prime powers

I am trying to implement an algorithm that can compute the square root of a quadratic residue mod a prime power. For integers $a$ such that $p\not\mid a$ $p\neq 2$ it's relatively straightforward ...
1
vote
3answers
877 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
votes
1answer
85 views

Most effective way to solve system of non-linear equations with unique set of roots

What is the most effective way to solve a system of non-linear equations if we know for sure that they have a unique set of roots?
0
votes
2answers
88 views

Find $\lim_{x \to \alpha}[1+ax^2+bx+c]^\frac{1}{x-\alpha}$

If $\alpha , \beta$ be the roots of $ax^2+bx+c=0$. Find $$\lim_{x \to \alpha}[1+ax^2+bx+c]^\frac{1}{x-\alpha}$$ Here $\alpha +\beta=-\frac{b}{a}$ and $\alpha \beta=\frac{c}{a}$. How can I ...
0
votes
0answers
80 views

Isolation of zeros in the case of univariate analytic functions expressed as a bivariate function.

We know that the zeros of an analytic non-constant function are always isolated. A proof is here. Let $L(v)$ be an analytic function in $v$, where $v\in\mathbb{R}$. Let us write $L(v) \equiv L(v,p)$ ...
0
votes
0answers
68 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 ...
0
votes
2answers
84 views

Finding roots of product of two polynomials

Let $P$ and $Q$ be polynomials of degree $2$ and $3$ respectively. If we know the roots of both $P$ and $Q$, is there an easier way of finding the roots of the product $PQ$? Do we really have to ...
1
vote
1answer
2k views

How does one prove that a polynomial has no rational roots in general?

How can we prove that a polynomial only has rational roots when we know the coefficients and the degree? For instance, in illustration, how would we show this for $x^8 ...
1
vote
0answers
55 views

Lower-bounding the distance between zeros of a continuous function

Consider a continuous function of the form: $L(v) = \sum_{i = 0}^{m}[vA_{i} - B_{i}]p^{i}$ where $p$ is the root of the polynomial equation: $vf(p) - g(p) = 0$ with $f(p)$ and $g(p)$ being two ...
1
vote
2answers
3k views

Convergence of Bisection method

I know how to prove the bound on the error after $k$ steps of the Bisection method. I.e. $$|\tau - x_{k}| \leq \left(\frac{1}{2}\right)^{k-1}|b-a|$$ where $a$ and $b$ are the starting points. But ...
0
votes
1answer
78 views

Iterarions count in Newton's method;

How many iterations must I do for getting $n$ signs after floating point in calculating square root by Newton's method P.S Sorry for my bad English. Please mention to me where I've done mistakes. ...
3
votes
1answer
204 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 ...
7
votes
3answers
244 views

Is the complex derivative “speed”?

The first thing I was told about the real derivative is that it's "how fast the function is growing" at a given point. This interpretation wasn't addressed in my complex analysis classes. Can the ...
11
votes
1answer
547 views

Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified?

I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained $$\begin{eqnarray} \frac{\sin z}z&=&\frac{\sin(x+iy)}{(x+iy)}\\ &=& ...
4
votes
1answer
570 views

How to solve an equation using Newton's method with and without backtracking?

Lets assume I have this equation: $$\log(e^x+e^{-x})=2x+5,\quad x \in (-50,50).$$ As always we have to pick a starting point to solve this by Newton's method, but how can i know for what initial ...