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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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Computing the root of a polynomial that has the lowest imaginary part

Suppose that we have a polynomial $P$ of degree $n$ whose roots are known to be all complex and with distinct imaginary parts. With such conditions, $P$ should have a unique root $z_0$ such as $|\Im(...
edrezen's user avatar
  • 243
-2 votes
0 answers
47 views

conjugate of x roots to rationalise a denominate not containing only 1 or 2 roots but as many as needed

I was wondering if anyone could help me explore this concept and give me guidance, to rationalize the denominator of a fraction involving roots in tis denominator we take the conjugate of its ...
Federico Ruck's user avatar
3 votes
1 answer
84 views

How to find principal value of the cubic root?

I tried to find principal value for $\sqrt[3]{z}$ , I started from $$ z=w^3 $$ So $$ w_1=\sqrt[3]{r} \exp\left(\frac{Arg(z)}{3} i\right)$$ $$ w_2=\sqrt[3]{r} \exp\left(\frac{Arg(z)+2\pi}{3} i\right)$$ ...
Faoler's user avatar
  • 1,637
6 votes
4 answers
114 views

$\frac{1}{x} + \frac{1}{x-1} + \ldots + \frac{1}{x-n} = 0$ has only one root $(0;1)$

Prove that: $$\frac{1}{x} + \frac{1}{x-1} + \ldots + \frac{1}{x-n} = 0$$ has only one real root in $(0;1)$ for all positive integer $n>1$ Here is what I tried: Rewrite the equation as: $$\frac{(x-1)...
Lục Trường Phát's user avatar
0 votes
1 answer
49 views

Finding $\delta$ for convergence of $f(x)=x^3+2x^2-3x-1$ at its approximate root.

I've had some exposure to elementary analysis and I am currently going through a problem in numerical analysis involving finding roots using Newton's method. The algorithm has convergence issues which ...
Mario Figueroa's user avatar
-1 votes
0 answers
51 views

Find the number of solution for an equation $f(x)=0$ we have $f(x)=3x^2+2ax+b\;$ and $\int_{-1}^{1}|f(x)|dx<2$ [closed]

We consider the function $f$ defined by $f(x)=3x^2+2ax+b\;$; $a$ and $b$ are two reals numbers. we have : $$\int_{-1}^{1}|f(x)|dx<2$$ The numbre of solution of the equation $f(x)=0$. chose the ...
user579102's user avatar
2 votes
2 answers
128 views

Find the maximum of $|a-b|$ if the equation $x^3-x^2+ax-b=0$ has real and positive roots. [closed]

This is an integer type question (round off to nearest integer) stating: "Find the maximum of $|a-b|$ ($a,b \in \mathbb R$) if the equation $x^3-x^2+ax-b=0$ has real and positive roots." My ...
Ritvik Bansal's user avatar
0 votes
1 answer
79 views

How do we differentiate the function $f(x) = (3x^{3} - 4x^{2} + 8x)\sqrt{6x^{2} + 3x}$? [closed]

I hope someone can help me with this derivative, I have already made this: At the first part, I did not have issues with the root derivative and then multiply it with $(3x^3-4x^2+8x)$, that part is ...
Gerardo Correa GCOES's user avatar
1 vote
3 answers
209 views

Can we *really* do algebraic operations involving roots on C?

With BSc in Maths and loads of grey hair, something has been on my mind for decades, and I couldn't quite enunciate it. Let me try. Root is inherently "multi-valued" operation. So $$ \sqrt{4}...
avloss's user avatar
  • 119
2 votes
1 answer
51 views

How to prove that a $k$-tuple root of an equation $f(x) = 0$ is a root of $f^{(k-1)} (x) = 0$ but not of $f^{(k)} (x) = 0$?

I am reviewing The Penguin Dictionary of Mathematics (4th edition, 2008), edited by David Nelson. In the section multiple root we have: For any equation $f(x) = 0$ a multiple root is also a root of ...
Prime Mover's user avatar
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3 votes
1 answer
97 views

Real roots of $x^4+ax^3+bx^2+cx+1=0,$ when $a,b,c$ are real and $b\ge\frac{a^2+c^2}{4}$

For real $a,b,c$ and $$b \ge \frac{a^2+c^2}{4}\tag{*}$$ the given polynomial equation $$f(x)=x^4+ax^3+bx^2+cx+1=0\tag{**}$$ can be re-written as $$f(x)=(x^2+ax/2)^2+(b-a^2/4-c^2/4)x^2+(cx/2+1)^2\ge 0\...
Z Ahmed's user avatar
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1 vote
0 answers
53 views

Does there exists something like the BKK Theorem for polynomials over finite fields?

I'm trying to count the number of common $\mathbb{F}_q^\times$-zeros of some polynomials $f,g \in \mathbb{F}_q[x,y]$. I thought I had a solution using the BKK theorem but I reread the theorem ...
Amelia Gibbs's user avatar
0 votes
0 answers
115 views

Prove that this limit is equal to $\sqrt{2}$ for the function $f(x)=x^2-2$ for an arbitrary seed point $s$.

Mathematica knows that: $$ s + \frac{1}{1-\lim_\limits{n\ \to\ \infty}\left[\frac{\displaystyle\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{f\left(k/n + s -1/n\right)}}{\displaystyle\sum _{k=1}^...
Mats Granvik's user avatar
  • 7,420
0 votes
1 answer
43 views

Help with Analytic Function Zero in Annulus

I'm working on a problem involving an analytic function in an annulus, and I need some help with the second part of the question. Here is the problem statement: Suppose that $ f(z) $ is analytic in ...
tree tree juice's user avatar
4 votes
0 answers
99 views

Number of distinct real roots of $1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+.....\frac{x^{2025}}{2025!}=0$ [duplicate]

Find the number of distinct real roots of $$P(x)=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+.....\frac{x^{2025}}{2025!}=0$$ My try: I tried to find critical points by setting $P'(x)=...
Ekaveera Gouribhatla's user avatar
0 votes
0 answers
46 views

A root finding problem

While solving a partial differential equation, I obtained the following function: $$ F(z)=(k^2 - z^2)J_1^2(z)-z^2J_0^2(z) \tag 1$$ where $k \in (1, \sqrt2)$ and $z$ is a complex variable. I need to ...
FriendlyNeighborhoodEngineer's user avatar
7 votes
3 answers
947 views

"Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?

I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
Hooman J's user avatar
  • 237
1 vote
1 answer
49 views

Asymptotic roots of equation involving Bessel functions

Let $r_2>r_1>0$, $g\ge 0$, and let $\lambda_n$, $n=1,2,\ldots,\infty$, be the positive roots of \begin{equation} J_g(xr_2)Y_g'(xr_1)-J_g'(xr_1)Y_g(xr_2)=0, \end{equation} where $J_g'(xr_1)$ is ...
Jog's user avatar
  • 369
1 vote
1 answer
35 views

Natural roots of multivariate integer polynomials

I know the problem of determining whether a multivariate integer polynomial has integer roots is in general undecidable. However, does this change when we consider only its natural roots?
qucchia's user avatar
  • 140
3 votes
1 answer
94 views

Closed form for $A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$?

Is there a closed form for $A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$ ?? We know $$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{\pi \ln( 27 + 5 \sqrt {29})}{\sqrt {...
mick's user avatar
  • 16.4k
5 votes
1 answer
423 views

Finding the real roots of an octic polynomial (degree eight). [duplicate]

I have been spending some time on a question which I encountered in the PYQs of a small maths competition from $2$ years ago. It is from the topic quadratic equations and polynomials, it however deals ...
Sarvagya's user avatar
1 vote
1 answer
47 views

Properties of Nth roots and fractional powers

Context: I'm programming an arbitrary precision math library and created some weird algorithms to calculate a number raised to non-integer powers due to optimizations. From my understanding, raising ...
jared soto's user avatar
0 votes
0 answers
23 views

number of zeros of a power series defined by an absolutely convergent sequence

Let $a = (a_n)_{n \in \mathbb{N}}$ be an absolutely convergent sequence such that each $a_n \in \mathbb{R}$, and define $f_a(x) = \sum_{n \in \mathbb{N}} a_n x^n$ for $x\in\mathbb{R}$. To avoid a ...
hs12's user avatar
  • 1
0 votes
1 answer
89 views

If $n$ is an even and $\alpha, \beta$ are the roots of the equation $x^2 + px + q = 0$ and also of the equation $x^{2n} + p^nx^n + q^n = 0$ ...

If $n$ is an even and $\alpha, \beta$ are the roots of the equation $x^2 + px + q = 0$ and also of the equation $x^{2n} + p^nx^n + q^n = 0$ and $f(x) = \frac{(1+x)^n}{1+x^n}$ where $\alpha^n + \beta^n ...
Ash_Blanc's user avatar
  • 1,133
1 vote
0 answers
87 views

Hard/Interesting analytical problem to solve: $0 = a_{1} \ln(1+a_{2}e^{a_{3}x^{2}+a_{4}x+a_{5}})+a_{6}x^{2}+a_{7}x+a_{8}$

Is there some trick or approach I'm missing in attempting to solve \ref{1}, or better \ref{2}, analytically. $$0 = a_{1}\ln(1+a_{2}e^{a_{3}x^{2}+a_{4}x+a_{5}})+a_{6}x^{2}+a_{7}x+a_{8} \tag{1}\label{1}$...
Mitternachtian's user avatar
0 votes
0 answers
53 views

Location of roots of polynomials: Understanding the proof

Assume a polynomial: $$p(z)=a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z^1+a_0$$ I am going throught the proof of the theorem that locates the roots of a polynomial inside an open ball. The theorem is as follows:...
mathCurious's user avatar
6 votes
2 answers
93 views

Optimal length of rope for sliding across a gap

I'm trying to solve a physics problem that I heard ~10 years ago in undergrad that was casually posed to me without a solution in mind; it has been bothering me ever since! Please let me know if this ...
pretzelKn0t's user avatar
1 vote
1 answer
62 views

Cosine as nested roots

I have been playing around with circles lately, and I have found an interesting limited relationship between prime factors and cosine. Have the form of: $$\cos{\left(2\pi\frac{1}{p}\right)}$$ And that ...
John Clement Husain's user avatar
1 vote
2 answers
89 views

Roots of $x^2+x+2$ over $\mathbb{Z}_3[X](i)$

As the polynomial $f(x)=x^2+x+2$ is irreducible, then $$ \mathbb{Z}_3[X]/\langle f(x) \rangle = \{p(x)+\langle f(x) \rangle \: : \: p(x) \in \mathbb{Z}_3[X]\} = \{ax+b+\langle f(x) \rangle \: : \: a,...
baristocrona's user avatar
6 votes
1 answer
217 views

Zeros of a reciprocal polynomial

Let $k$ be a positive even integer and consider the polynomial $$ f_k(z)=z^k(z+1)^k+z^k+(z+1)^k. $$ Numerical computation suggests that the zeros of this polynomial behave as follows: If $\mbox{Re}(z)\...
Itachi's user avatar
  • 656
0 votes
3 answers
90 views

Can we find the root of this equation

Give an equation below: $$ \frac{x^k-a^k}{x-a}=c \qquad (1) $$ where $1<a<x$, $0<k<1$, and $c>0$. I can easily find the numerical root of (1) by using Newton's method or the other tools....
Tyke's user avatar
  • 159
0 votes
0 answers
32 views

Prove that a trigonometric polynomial has 2n roots [duplicate]

$0<a_0<a_1<a_2<...<a_n$ Prove:$$a_0+a_1\cos\theta+...+a_n\cos n\theta $$ has 2n distinct roots in $(0,2\pi)$ This is a question in my textbook,the author leaves a hint.That is we should ...
MathNoob's user avatar
  • 331
1 vote
0 answers
57 views

Simplifying two rational expressions with roots

Can you please check the steps I followed to simplify the following two expressions? Given that $0 < n < 1$, simplify: $$\left( \frac{\sqrt{1+n}}{\sqrt{1+n} - \sqrt{1-n}} + \frac{1-n}{\sqrt{1-n^...
Aleksandar Živković's user avatar
0 votes
0 answers
76 views

Can the roots of two cubic apolar polynomials, each having only real roots, ever be interlaced?

Consider two cubic real polynomials $f(x)$ and $g(x)$, each of them having only real roots. If we further assume that $f(x)$ and $g(x)$ are apolar, can their roots ever be interlaced? I have a feeling ...
Malkoun's user avatar
  • 5,375
3 votes
1 answer
196 views

About a function bounded by two polynomials

Let $P(x)=x^n+\Sigma_{k=0}^{n-1}a_kx^k $ and $Q(x)=x^n+\Sigma_{k=0}^{n-1}b_kx^k $ be polynomials with real coefficients such that $n\ge 4$ is even and $a_{n-1}<b_{n-1} $ . Let $f(x)$ be a function ...
user-492177's user avatar
  • 2,589
1 vote
1 answer
42 views

Quadratic equation with a relation between its coefficients

Given the quadratic equation $ax^2+bx+c=0$, where $a, b, c \in\mathbb{R}$ such that $4(a+b)+7c=0$, $(a\neq0)$ prove that: All of the quadratic's roots are real. Atleast one of the roots is in the $[0,...
fikooo's user avatar
  • 409
3 votes
2 answers
212 views

Generalization of Integer-Powered Sums Problem

I am trying to solve a problem that involves the sum of the $n$-th roots of positive reals. Specifically, the task is to determine all sets of positive reals $a_1, a_2, a_3$ such that $\sqrt[n]{a_1}+\...
Snowball's user avatar
  • 1,023
1 vote
0 answers
41 views

Is there a Relation for Exponentiation Similar to $\leq$ or $\backslash$ (divisibility)?

I'm trying to define common mathematical options on the natural numbers. I am doing this because operations like subtraction are commonly constructed as just the addition of the inverse, however this ...
Isaac Sechslingloff's user avatar
0 votes
0 answers
28 views

Finding roots of the derivative of function with variable powers

I have this function $f(x) = r_\text{o1}\cdot\left(1-\left(\dfrac{r_\text{i1}}{x+r_\text{i1}}\right)^\frac{w_\text{i1}}{w_\text{o1}}\right)$. I also have $g(x)$ with the same structure but instead of ...
Ivan's user avatar
  • 113
4 votes
1 answer
123 views

Polynomial maximizing its discriminant

Consider a polynomial $p(x) = (x-x_0)(x-x_1)\dots (x-x_{n-1})(x-x_n)$, with $-1 = x_0 < x_1 < \dots < x_{n-1} < x_n = 1$. What values for the roots $x_1, \dots x_{n-1}$ maximize the ...
Marc Alexa's user avatar
4 votes
2 answers
119 views

Questions on integrating $\int_0^{\infty} \frac{x^{\frac{1}{3}}}{1 + x^2} dx$ using contour integrals

We have the following integral to solve $$ I = \int_0^{\infty} \frac{x^{\frac{1}{3}}}{1 + x^2} dx $$ I've managed to solve this integral. Using the substitution $u = x^2$ we can show that $I = \frac{1}...
Noud's user avatar
  • 533
1 vote
1 answer
41 views

Can a non trivial computable function have an uncomputable root

Can a non trivial computable function have an uncomputable root, and how would you show this. Formally, Given $f$ a computable function and $r$ a real uncomputable number. Then $f(r) \neq 0 \vee ( \...
Sam Coutteau's user avatar
1 vote
0 answers
20 views

Root finding of multivariate polynomials over the integers

TL;DR: is there any library for multivariate polynomial root finding over the integers? I'm trying to implement an attack on RSA with known bits of p by using Coppersmith, such as shown in this paper. ...
Cnoob's user avatar
  • 11
7 votes
3 answers
244 views

constraints on the sum and product of roots of quadratic equation assuming less than unity roots

I am solving a math contest problem. Assume we have the quadratic equation $x^2+a_1x+a_2=0$ where $a_1,a_2\in \mathbb{R}$ are real numbers. The roots of this equation can be found as (from equation it ...
K.K.McDonald's user avatar
  • 3,263
16 votes
1 answer
412 views

Why do the roots of $\int_0^x (1-s^2)^n ds$ lie on a lemniscate?

Consider the polynomial $$f_n(x)=\int_0^x (1-s^2)^n ds$$ for some integer $n$. I am interested in these polynomials for reasons unrelated to this question: these are the odd polynomials of which the ...
Wouter's user avatar
  • 7,853
0 votes
1 answer
60 views

There is no polynomial p(x) for which there is a single line that is tangent to the graph of p(x) at exactly 100 points.

False. For p(x) with exactly 100 multiple roots the X-axis is such a line. (This is essentially the only way: if y = ax + b is such a line for a polynomial q(x), then the X-axis is such a line for the ...
Ayush tripathi's user avatar
3 votes
2 answers
156 views

Location of roots of $z+z^2/2+\cdots+z^n/n$

I am interested in the complex roots of $P_n(z)=\sum_{k=1}^n\frac{z^k}{k}$. With the help of some computer algebra system, I have been able to see that, with the exception of the root $0$, all the ...
Adren's user avatar
  • 7,602
1 vote
0 answers
71 views

Recurrence-defined polynomial with purely imaginary roots

I am working on a polynomial of arbitrary size, described by the recurrence relation: $$P_{n+2}=-xP_{n+1}+(n+1)(n+2)P_n, \text{ with } P_0=1, P_1=-x. $$ I observe that for $n$ being odd, $P_n$ has one ...
Ba_nanza's user avatar
  • 138
0 votes
0 answers
70 views

the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $

I was looking at the zero's of $$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$ for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$. Now this clearly relates to the Riemann zeta: $$f(s,a) + \...
mick's user avatar
  • 16.4k
4 votes
1 answer
445 views

What is the difference between roots and zeroes? [duplicate]

Suppose I have a polynomial of degree 6. It crosses the x-axis at 3 distinct points, and the graph of the polynomial touches the x-axis at one of those 3 points (a repeated root). Question 1: What is ...
Kampann's user avatar
  • 117

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