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 (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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10
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68 views

Property of a polynomial with no positive real roots

The following is an exercise (Exercise #3 (a), Chapter 3, page 28) from Richard Stanley's Algebraic Combinatorics. Let $P(x)$ be a nonzero polynomial with real coefficients. Show that the ...
10
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171 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
8
votes
0answers
163 views

Let $x_n$ be the (unique) root of $\Delta f_n(x)=0$. Then $\Delta x_n\to 1$

Note that by Cesaro's Theorem, we have as a consequence $$\frac{x_n}n\to 1$$ Consider $$r_n(x)=e^{-x}-\sum_{k=0}^n (-1)^k\frac{x^k}{k!}$$ and $$f_n(x)=(-1)^{n+1}e^{-x}r_n(x)$$ One can argue by ...
6
votes
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57 views

Can we express the roots of all polynomials in terms of roots of some special polynomials?

We can describe the roots of quadratic equations in terms of addition, subtraction, multiplication, division, and the square-root function $\sqrt a$ which computes a root of the special polynomial ...
6
votes
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157 views

Generating Functions, Recursive Polynomials

At the CMFT international conference in Turkey (2009), the following open problem was given: Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
5
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59 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, ...
5
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0answers
122 views

Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry ...
5
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91 views

Can even degree Legendre polynomials have roots in common?

I'm wondering whether the Legendre polynomials $P_m(x)$ and $P_{m+2k}(x)$, with $m$ even and $k \in \mathbb{N}^+$, can have common roots. For $k=1$ it is straightforward to show that there are no ...
5
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58 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
5
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80 views

How does the polynomial transformation $P(x) \mapsto P(x) + c$ alter the roots of that polynomial? Specifics inside.

Consider a real quadratic polynomial $Q_k(x) = (x-\nu)(x-\omega_k) - g_k^2$. I can interpret $Q_k(x)$ as a translation of the polynomial $$ (x-\nu)(x-\omega_k) = ...
4
votes
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151 views

Find the complex (or real) roots of $e^{\frac{3 x}{2}}+2 \cos \left(\frac{\sqrt{3} x}{2}\right)$

Define for natural $n\geq 2$ $$G(x,n)= \sum _{k=0}^\infty \frac{x^{k n}}{(k n)!}= \frac{\sum _{k=0}^{n-1} e^{x e^{\frac{2 i \pi k}{n}}}}{n}= G(x e^{\frac{2 i \pi}{n}},n)= \prod_{m=1}^\infty ...
4
votes
0answers
51 views

Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
4
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57 views

Roots of a polynomial that is composed n times with itself

Let $f(x)=x(4x^2-3)(64x^6-96x^4+36x^2-3)$ and $f^{(n)}=f(f(f(\cdots f(x))\cdots)$ (composed with itself $n$ times). Prove that for all positive integers $n$, $f^{(n)}(x)=x$ has $9^n$ distinct ...
4
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105 views

Roots of a polynomial

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then ...
4
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291 views

Prove equation has only one root in a specific interval

Prove that the following equation has only one solution in the interval $[-\text{min}(a_i), +\infty]$: $f(x) = \left(\sum_{i=1}^n \frac{1}{a_i + x}\right)\times \left(\sum_{i=1}^n \frac{a_i b_i}{(a_i ...
4
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62 views

On the location of the roots of a polynomial

Consider the following two polynomials \begin{align} p(s)&:=s^n+\alpha_{n-1}s^{n-1}+\cdots+\alpha_1s+\alpha_0,\\ q(s)&:=s^{n-1}+\alpha_{n-1}s^{n-2}+\cdots+\alpha_2 s+\alpha_1, \end{align} ...
4
votes
0answers
123 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ ...
4
votes
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140 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 ...
4
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544 views

Determine the number of zero points of $z^8-5z^3+z-2$ within the open unit circle (Rouché?)

How many zero points does the polynomial $z^8-5z^3+z-2$ have within the open unit circle? Hello, consider $$ \gamma\colon [0,2\pi]\to\mathbb{C}, \varphi\longmapsto\exp(i\varphi) $$ and ...
4
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0answers
533 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
3
votes
0answers
53 views

Why does Ridders' method work as well as it does?

I've just read section 9.2.1 in Numerical Recipes Ed. 3 (Press et al. 2007), which describes Ridders' method of root finding. I understand that allowing for some curvature of the function by ...
3
votes
0answers
43 views

What is the number of zeros of antiderivatives of $(x-1)(x-2)^2(x-3)^3(x-4)^4$?

For each $x \in \mathbb{R}$, let $f(x) = (x-1)(x-2)^2(x-3)^3(x-4)^4$. This defines a function $f : \mathbb{R} \to \mathbb{R}$. There is a unique natural number $k$ such that every antiderivative of ...
3
votes
0answers
107 views

Finding exact roots

I know of the rational root theorem to find all rational zeros and Newtons method of approximating zeros, but what if all the solutions are irrational/imaginary and you need exact answers for the ...
3
votes
0answers
62 views

Sum of zeros of $P(x)$

I asked this question here before too, but vaguely, hopefully, this time will be a better attempt: There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero ...
3
votes
0answers
38 views

Squares of finite fields (mod p*q)

Lets say we have $\mathbb{Z}_p$, where p is prime. For each element(x) we have two squares(y) so that $y^2=x$ ie if $p=7$ for $x=4$ we have $y_1=2,y_2=7-2=5,y=\pm2 $ ok, lets have ...
3
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67 views

Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
3
votes
0answers
147 views

On polynomials over finite fields?

Pick prime $q\in\Bbb Z$ such that $q>B^{3t}>(mB)^{2t}$. Suppose I have a multilinear polynomial $g(x)\in \Bbb Z[x_1,\dots,x_n]$ of degree $t$ with $m^t$ non-zero coefficients that bound by ...
3
votes
0answers
49 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
3
votes
0answers
54 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
3
votes
0answers
75 views

How can I find my eccentricity (k, of the incomplete elliptic integral of the second kind) using a binomial series or root-finding algorithm

My main objective is to rearrange the following to find $A=$. I start with $C=\int^{B}_{0}\sqrt {1+\dfrac {A^{2}}{B^{2}}\cos^{2}\left(\dfrac {x}{B}\right) }dx$ By substituting $y=x/B$ and using ...
3
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69 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
3
votes
0answers
57 views

Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
3
votes
0answers
381 views

Trigonometric functions of angle fractions

I've just encountered a problem that seems to me interesting enough so that some result exists on the subject. I was working on a problem in complex analysis, in which I needed the fifth root of a ...
3
votes
0answers
78 views

Roots of the derivative as symmetric (?) functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
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 ...
3
votes
0answers
66 views

Existence of roots of a polynomial equation when coefficients have varying weights

I have two $n-$degree polynomials $f_{1}(p)$ and $f_{2}(p)$, where the domain of $p\in[0,1]$. I know that $\exists$ $0 < p_{1} < 1$ such that: $f_{1}(p_{1}) = f_{2}(p_{1})$. Let ...
3
votes
0answers
38 views

Study a particular polynomial sequence

Let us define the following sequence of polynomials for every two non-negative integers $i,d$: $$s_i^d(w)=\sum_{j=0}^{d+1} (-1)^j {d+1\choose j} (j+1)^i w^{d+1-j}.$$ Conjecture: The sequence ...
3
votes
0answers
50 views

Existence of a Root of Elementary Monomials

Let $m_\lambda(X_1(t),X_2(t),...X_N(t))$ be a monomial symmetric function with partition $\lambda$. For example: $$ m_{(3,1,1)}(X_1(t),X_2(t),...X_N(t)) =X_1^3X_2X_3 + X_1X_2^3X_3 + X_1X_2X_3^3 $$ ...
3
votes
0answers
842 views

Existence of n distinct (real) roots of an orthogonal polynomial

I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof ...
2
votes
0answers
48 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
2
votes
0answers
30 views

Finding the roots of a polynomial with limited information

Let $\ f(x) \in \mathbb{R}[x]$ be a 7-th degree polynomial, such that. $$ f(0)=0 \land f(i)=-3i $$ $$ f'(0)=0 \land f'(i)=-21 $$ Find all the complex roots of $ f(x)-3x^7$. Find all possible ...
2
votes
0answers
39 views

Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Nutch arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow ...
2
votes
0answers
37 views

Find all such $a$ that $x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|$ has at least one root.

In the equation, $a$ is a parameter and $x$ is a variable: $$x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|.$$ I want to find all values of $a$ that make the equation have at least one ...
2
votes
0answers
43 views

Number of “possible” rational roots of a univariate polynomial

Is it possible to determine the number of possible rational roots of a single variable polynomial? polynomial a0 + a1x + a2x^2 + ... + anx^n. This is to find ...
2
votes
0answers
45 views

Zeros of real analytic functions

It is known that given a discrete set of points on the complex plane with prescribed multiplicities, there exists a complex holomorphic function with zeros at those points with the prescribed ...
2
votes
0answers
29 views

Roots of polynomials

Question: The equation $x^3 -x^2 - 3x - 10 = 0$ has roots $\alpha,\,\beta,\,\gamma$ Let $u = -\alpha + \beta + \gamma$. Show that $u + 2\alpha = 1$, and hence find a cubic equation having roots ...
2
votes
0answers
100 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
2
votes
0answers
42 views

Find cubic equation roots: what choice to make for cube root to avoid circularly permuting the roots?

My work involves solving a cubic equation similar to this $a x^3+b x^2+c x +d$ which has three roots, $x_1, x_2, x_3$ (as given in https://en.wikipedia.org/wiki/Cubic_function.) ...
2
votes
0answers
48 views

Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
2
votes
0answers
47 views

Finding roots of $4$th degree conjugate reciprocal polynomial

I am developing a computer program and the following polynomial, of which I need to obtain the roots, turned up $$Ax^4 + Bx^3 + Cx^2 + \overline{B}x + \overline{A}, \quad \text{where } A, B,x \in ...