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
votes
2answers
94 views

Polynomial whose roots are very nearly integers

Let $n\geq 6$ be an integer ; consider the polynomial $$ P_n=\big((x-1)(x-2)\ldots (x-n)\big)^2+(n+1) $$ Let $\varepsilon=\frac{n^4}{173\big((\lfloor \frac{n}{2} \rfloor)!\big)^2}$. I have checked ...
0
votes
0answers
36 views

roots of a polynom in a localization of a UFD

let $ {R} $ be a UFD, $ Q $ the localization of $ R $. I need to find all the roots in $ Q[i] $ of the polynom: $ f(x) = x^4 + \frac {4} {5+i}x^3 - \frac {6+10i} {2+3i}x^2 - \frac {12} {5+i}x + \frac ...
0
votes
2answers
43 views

Roots of a given equation

How can I show that the equation $$e^x-\ln(x)-2^{2014}=0$$ has exactly two positive roots?
5
votes
1answer
86 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
2
votes
2answers
71 views

Question about bisection method

We have $f(x)=(x-1)^3(x-2)(x-3)$. $a_0<1,b_0>3$. We had to show that if $\frac{a_0+b_0}{2}\ne 1,2,3$, there is one root of $f$ that we can't get it by the bisection method. I guess that this is ...
1
vote
1answer
84 views

Rational Root theorem issue

I've given my class an example: $$2x^3+3x^2+6x+4=0$$ By the rational root theorem, if there is a rational root then it should be of the form $\frac{p}{q}$ where $p$ is a factor of 4 and $q$ is a ...
1
vote
2answers
35 views

Lemma 2.5.5 Boas, Entire functions

I'm reading Boas, Entire functions, but I don't understand lemma 2.5.5, which states that $\sum_{1}^{+\infty}\frac{1}{r_{n}^{\alpha}}$ and the integral $\int_{0}^{+\infty}t^{-\alpha -1}n(t)dt$ ...
2
votes
1answer
66 views

Order of growth of $ \prod_{n=1}^{+\infty} (1-e^{-2\pi n}\cdot e^{2\pi i z})$

The order of an entire function $f$ id defined as $$ord ( f) = inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ I have $F(z) = ...
1
vote
1answer
43 views

A limit involving the exponent of convergence

Let $f$ be an entire non-constant function with at least one zero. If $\{z_{j}\}_{j\in \mathbb{N}}$ are the zeros of $f$, set $$b =\inf\left\{\lambda >0 \ | \sum_{j}\frac{1}{|z_{j}|^{\lambda}}< ...
3
votes
3answers
170 views

$\alpha,\beta,\gamma$ are roots of cubic equation $x^3+4x-1=0$

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$ and $\displaystyle \frac{1}{\alpha+1},\frac{1}{\beta+1},\frac{1}{\gamma+1}$ are the roots of the equation $\displaystyle ...
0
votes
1answer
22 views

primitive roots, field dimension

Let $\zeta$ be a primitive $m$-th root of $1$. Determine the values of $m$ such that: $[\mathbb Q$($\zeta$):$\mathbb Q$]$=2$. The only thing I have in mind is that $[\mathbb Q $($\zeta$):$\mathbb ...
0
votes
1answer
41 views

Rate of Convergence of Generalized Iterative Method

Consider the generalized iterative method for finding polynomial roots: $z_{k+1}=z_k +d\frac{(1/p)^{(d-1)}(z_k)}{(1/p)^{(d)}(z_k)}$ where d is a positive integer. Note that Newton's Method is a ...
6
votes
2answers
127 views

Prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$

I need to prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$ My attempt: Let $a=\sqrt[2012]{2012!}$ and $b=\sqrt[2013]{2013!}$ Then $\displaystyle\frac{b^{2012}}{a^{2012}}=\frac{2013}{b}$ ...
2
votes
2answers
57 views

rectangle where $\cos{z} =iz$ has exactly one solution

Determine a rectangle inside which there is exactly one solution of the equation $\cos{z} = iz$. I know the following result: Let $f$ be holomorphic in $\Omega$ with $a \in \Omega$. Let $f(a)= b$ is ...
0
votes
4answers
80 views

On the roots of a polynomial [closed]

Let $f(x)= x^3 - 3x + 1$. Show that if a complex number $a$ is root of $f(x)$, then $a^2-2$ is too.
0
votes
2answers
113 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
10
votes
4answers
162 views

Coefficients of a polynomial also are the roots of the polynomial?

How many real solutions $(r_1, r_2, \cdots, r_n)$ are there such that $(r_1, r_2, \cdots, r_n)$ are the roots of the polynomials $x^{n} + r_1 x^{n-1} + r_2 x^{n-2} + \cdots + r_n$ For $n = 2, 3, 4$ I ...
6
votes
3answers
140 views

Number of integral roots of a polynomial

Let $p(x)$ be a polynomial with integral coefficients. Let $a$, $b$, $c$ be three distinct integers such that $p(a) = p(b) = p(c) = -1$. Find the number of integral roots of $p(x)$.
2
votes
1answer
46 views

uniformly bounded sequence of non constant holomorphic functions

Let $\{f_n\}_{n=1}^{\infty}$ be a uniformly bounded sequence of nonconstant holomorphic functions in a connected open set $\Omega$. Let $f \not \equiv 0 $ be a holomorphic function in $\Omega$. ...
0
votes
3answers
64 views

Absolut value of cubic polynomial roots lower than 1

Assume we have a cubic polynomial $ x^3 +bx^2+xc+d=0 $, with b,c,d real numbers. Let $x_1, x_2, x_3 $ be the roots, either real or complex. What is the relation of the coefficients b,c and d in ...
4
votes
2answers
79 views

meromorphic function in the unit disc with only one pole of order n

Let $f$ be meromorphic in a neighborhood of $\{|z| \leq 1\}\setminus \{1/2\}$ and have a pole or order $n$ at $1/2$. Suppose that $|f| < 3$ on $\{|z|=1\}$. Show that for any $\phi \in \mathbb{R}$, ...
2
votes
3answers
85 views

theory of equations finding roots from given polynomial

If the equation $x^4-4x^3+ax^2+bx+1=0$ has four positive roots then $a=\,?$ and $b=\,?$ $\textbf{A.}\,6,-4$ $\textbf{B.}\,-6,4$ $\textbf{C.}\,6,4$ $\textbf{D.}\,-6,-4$ we can ...
1
vote
1answer
63 views

Find the root of the polynomial?

Consider the root of the polynomial $p(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_1x -1$. Suppose that $p(x)$ has no roots in the open unit disc in a complex plane and $p(-1)=0$. Show that ...
4
votes
1answer
53 views

About the zeros of $f_n(z)=\sum_{k=1}^n k^{-z}$.

Let $z$ be a complex number. Consider $f_n(z)=\sum_{k=1}^n k^{-z}$. Now I wonder : Are there infinitely many positive integer $n$ such that there exists a $z$ with $f_n(z)=0$ and $Re(z)>1$ ? I ...
0
votes
1answer
24 views

Repeated Eigenvalues in Systems of ODEs

Question is to find the general solution of the given system of equations below. $$ x' =\left(\begin{array}{rr}\frac{-3}{2} & \frac{-1}{4} \\ 1 & \frac{-1}{2}\end{array}\right)x $$ My ...
0
votes
1answer
87 views

Solve : $x^4 + 6x^3 -3x^2 + 2 = 0$

$x^4 + 6x^3 -3x^2 + 2 = 0$ To find the zeros, I tried this by Ferrari's method but got stuck where a value of 'lambda' has to be obtained.
13
votes
1answer
128 views

Prove $\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0$ has $n$ negative roots

Let's $n \in \mathbb{Z^+}$, how to $\text{prove}|\text{disprove}$ that: the equation $\boxed{\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0}$ has exactly $n$ distinct negative roots. My friend get ...
4
votes
2answers
60 views

Given $f \in \mathbb{Q}[x]$ irreducible. How many and which roots of $f$ are contained in $\mathbb{Q}[x]/(f)$?

It is a fact that struggle me for a while. When working with irreducible polynomial over $\mathbb{Q}$ it is natural to build the extension ${\mathbb{Q}[x]}/{(f)} $ in which "lives " one root of the ...
3
votes
2answers
54 views

Existence of holomorphic function with a sequence of zeros in the unit disc

The question is : Prove that there exists a holomorphic function $f$ on the open unit disc $\{z \in \mathbb{C} : |z| <1\}$ with the properties that $f(0) = 0$ and $f(1-1/n)=1$ for every integer $n$ ...
0
votes
2answers
60 views

A 3rd degree polynomial $P(x)$ has three unequal real roots. What is the least possible # of unequal real roots for $P(x^2)$

I got that if P(x) is a 3rd degree polynomial then P($x^2$) must be a 6th degree polynomial. I don't know how to proceed from here.
2
votes
4answers
115 views

Find the solution of the equation

Find all real solutions of this equation : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
1
vote
1answer
52 views

Solution of equations of the form: $a^x+b^x+c=0$

Is it possible to solve equations of the form: $a^x+b^x+c=0,\;abc\neq0$ with analytical methods; if so, how is this done?
0
votes
0answers
51 views

Demostrating that a trascendental function can be expressed as a product of its roots [duplicate]

I'm doing a work about the product of Wallis, a formula to calculate pi that is π/2= ∏ (2n/2n-1)· (2n/2n+1) from n=1 to n=∞. I have to prove the formula, and I have been searching in books and all ...
3
votes
1answer
112 views

zeros of a function holomorphic in the closed unit disc

Let $f$ be a holomorphic function in a neighborhood of the closed unit disc $\{z \in \mathbb{C} : |z| \leq 1\}$, and suppose that $\Re{(\bar{z}f(z))} > 0 $ when $|z| = 1$. Prove that $f$ has ...
2
votes
3answers
98 views

How bad, really, is the bisection method?

We know that the bisection method for root finding is slow (linear convergence), but has the advantage of always working for a continuous function, if we start with a interval which brackets the root. ...
1
vote
1answer
72 views

How to calculate the integral of a function with a root?

I have to solve this integral: $\int\limits_{-1}^1(3x^3-5x^2+12x-9)~dx$ I used Grapher (a nifty program that comes with Mac OS X) to display the curve of $f(x)=3x^3-5x^2+12x-9$ and it obviously has ...
4
votes
1answer
136 views

Roots of $f(x)=\sin(x)-ax$

How many roots are there of the function $f(x)=\sin(x)-ax$, where $a$ is a positive number? Clearly for all $a$, $x=0$ is a root; if $a>1$ that is the only root. The roots will also be symmetric ...
7
votes
5answers
201 views

Analytical solution to $a^x+b^x=x$

Maybe stupid question, but I am wondering. Is there an analytical solution to equation $$a^x+b^x=x$$ for general $a$, $b$. How should I tackle this problem, if I want to find at least one $x$. ...
3
votes
0answers
112 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 ...
0
votes
1answer
58 views

Algebraically find roots of a function composed of linear equations and trigonometric functions

I have the following equation of $t$: $\text{C0}+(\text{C1}+\text{C2} t) \cos (\text{C4} t)+\sin (\text{C4} t) (\text{C7}+\text{C8} t)+\text{C5} \cos (\text{C6} t)+\text{C9} \sin (\text{C6} t)=0$ ...
3
votes
2answers
82 views

What's the non-trivial root of $\lim \limits_{n\to \infty}\left(\sum_{k=0}^n x^{2^k}\right)^n$?

$$ \lim_{n\to \infty}\left(\sum_{k=0}^n x^{2^k}\right)^n=0 $$ always seems to have two real solutions. One trivial $x_0=0$ and another around $x_1=-0.65862...$ (see W|A @ $n=13$). Where does this ...
2
votes
2answers
31 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
4
votes
2answers
83 views

Use $\alpha, \beta, \gamma $ roots of a polynomial to construct another polynomial [duplicate]

Let $\alpha, \beta, \gamma $ be roots $\in \mathbb{C}$ of $x^3-3x+1$. Determinate a monic polynomial, degree $3$, witch roots are $1- \alpha^{-1},1-\beta^{-1},1-\gamma^{-1}$ The catch is that i can't ...
4
votes
0answers
86 views

Number of zeros equal number of linearly independent analytic functions

I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function $f(z)$ which have to satisfy the following boundary conditions: ...
2
votes
1answer
48 views

how to prove roots quadratics

the quadratic equation $3(k+2)x^2+(k+5)x+k=0$ has real roots show $(k-1)(11k+25) \geq 0 $ If $\Delta$ greater than $0$ it has real roots so, $$\Delta = (k+5)^2 - 4 \cdot (3(k+2))\cdot k$$ ...
0
votes
1answer
79 views

If f is n-times differentiable, and $f^n$ is never 0, then f has at most n zeros in R

Let $n \ge 0$, let $f:\mathbb{R} \rightarrow \mathbb{R}$ be n-times diff erentiable on $\mathbb{R}$, and assume that $f^{(n)}(x) \neq 0$ for all $x \in \mathbb{R}$. Show that $f$ has at most $n$ zeros ...
0
votes
2answers
114 views

Polynomial divisibility

Given $p(x) \in \mathbb Q[x] $ an irreducible polynomial, and $\alpha \in\mathbb C $ root of $p(x)$. Prove that if $q(x) \in \mathbb Q[x]$ it's a polynomial, such $q(\alpha) = 0$ then $p(x) \mid ...
4
votes
1answer
73 views

Solution of $Ax^5+Bx^3=C$

I have to find the positive solution of the type $Ax^5+Bx^3=C (A,B,C>0)$. It is well known that a polynomial of degree greater than $4$ does not admit an expression for the roots but I hope :D In ...
0
votes
2answers
96 views

Use a given zero to write P(x) as a product of linear and irreducible quadratic factors

The polynomial in question is: $x^4 - 8x^3 - 19x^2 + 288x - 612$ and the zero is $4 - i$. What I don't understand is how to go from the given zero to factorizing, especially as it's imaginary. ...
1
vote
0answers
64 views

How to prove that there are $O(T\ln T)$ zeros in the critical strip of the Riemann zeta function?

Define $F(T)$ as the number of solutions to $\zeta(a+ ti) =0$ for $0\le t\le T$ and $0<a<1$. How to show that $F(T)= O(T\ln T)$? For clarity, $\zeta$ is the Riemann zeta function, $i$ is the ...