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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Positive, Real Roots of Bivariate Polynomial
I have a question regarding lemma 3.1 in this paper. The lemma in question is as follows
Consider the function $f(x, \lambda) = ax^3 + bx^2 + cx + d$ where $a > 0$ is
fixed but for which the ...
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On how many circles will the roots of these constructed palindromic polynomials sit on?
The approximation to the Dirichlet eta function plus its conjugate:
$$\left(1-\frac{1}{2^{a+i b-1}}\right) \zeta (a+b i)+\left(1-\frac{1}{2^{a-i b-1}}\right) \zeta (a-b i)=\frac{1}{1^{a-i b}}+\frac{1}{...
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Number of real roots of a specific polynomial
I am working with the following polynomial:
$f_n := (x^2 + 2(n − 1)) \cdot (x − 2) \cdot (x − 4) \cdot (x-6) \cdots (x − 2(n − 2)) − 2$; $n\ge3$
I had to show that $f_n$ has exactly $n-2$ real roots.
...
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Does the Gauss-Lucas theorem generalize to arbitrary entire functions with a finite number of zeros?
The Gauss-Lucas theorem states that if $P(z)$ is a non-constant polynomial with complex coefficients, then all the zeros of $P'(z)$ lie within the convex hull of the set of zeros of $P(z)$.
Does this ...
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Computing the root of a polynomial that has the lowest imaginary part
Suppose that we have a polynomial $P$ of degree $n$ whose roots are known to be all complex and with distinct imaginary parts.
With such conditions, $P$ should have a unique root $z_0$ such as
$|\Im(...
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How to find principal value of the cubic root?
I tried to find principal value for $\sqrt[3]{z}$ , I started from
$$ 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)$$
...
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$\frac{1}{x} + \frac{1}{x-1} + \ldots + \frac{1}{x-n} = 0$ has only one root $(0;1)$
Prove that: $$\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)...
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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 ...
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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 ...
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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 ...
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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
$$
\sqrt{4}...
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How to prove that a $k$-tuple root of an equation $f(x) = 0$ is a root of $f^{(k-1)} (x) = 0$ but not of $f^{(k)} (x) = 0$?
I am reviewing The Penguin Dictionary of Mathematics (4th edition, 2008), edited by David Nelson.
In the section multiple root we have:
For any equation $f(x) = 0$ a multiple root is also a root of ...
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Real roots of $x^4+ax^3+bx^2+cx+1=0,$ when $a,b,c$ are real and $b\ge\frac{a^2+c^2}{4}$
For real $a,b,c$ and
$$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\...
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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 ...
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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}^...
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