1
vote
1answer
41 views

Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
0
votes
2answers
35 views

Real polynomials, complex zeroes and the Intermediate value theorem

I have a second grad polynomial p(x). For arguments sake lets say $$p(x) = x^2 + 16x + 76$$ I also have an inequation $$p(x) > 0$$ Now the inequation does not have a real solution, but only ...
0
votes
2answers
44 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
2
votes
1answer
56 views

What is a rational way of factoring polynomials?

Consider the following polynomial $$P_x:=x^4+1$$ I want to represent it as a product of two polynomials with real coefficients of grade $2$. I have done this using, call it, a brute force method. ...
11
votes
1answer
126 views

Will this sequence of polynomials converge to a Hermite polynomial pointwise?

While trying to solve this question my testing lead to an observation that I found interesting in its own right. Consider the linear transformation $L:P\to P$ from the space of polynomial functions ...
3
votes
1answer
80 views

Proving $\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$

I wish to prove the following inequality for $x\ne 0$: $$\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$$ Using the fact that I already prove: $$\cos x > 1 - \frac{x^2}{2}$$ My try: $\cos x = 1 - ...
30
votes
4answers
847 views

$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
2
votes
0answers
25 views

Show that $P_a(z)=0$ iff $z=N(a)$ for polynomial $P_a$

Let for $a_0=(a_0,a_1,...,a_n)\in\mathbb C^{n+1}$ the polynomial $P_{a_0}=\sum_{k=0}^na_kz^k$ and $z_0\in\mathbb C$ with $P_{a_0}(z_o)=0$ and $DP_{a_o}$ (the differential matrix) invertible. Show ...
4
votes
2answers
218 views

Show polynomial function is infinitely differentialble

Let for $\alpha=(\alpha_0,...,\alpha_n)\in\mathbb{C}^{n+1}$ the polynomial $p_\alpha :\mathbb{C}\to\mathbb{C}$ be given by $p_\alpha(z)=\sum_{k=0}^n \alpha_kz^k$. Show that with the identification ...
0
votes
1answer
64 views

Given a polynomial of degree 5, get minimum and maximum without using derivatives

Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial ...
2
votes
1answer
252 views

How to make this polynomial the zero polynomial?(recursively)?

Given a fixed $\beta \in \mathbb{R}$, I want to find the $c_0,...,c_n$ for arbitrary $n \in \mathbb{N}$ such that the polynomial \begin{align}P_n(z):=z(1-z) ...
1
vote
1answer
95 views

Root of order k

Im trying to show that a number $x_0$ a root of order $k$ of the polynomial $p(x)$ if and only if $p(x_0)=p^{'}(x_0)= ... =p^{(k-1)}(x_0)=0$ and $p^k(x_0)\neq0$. Is there an easy way to do this using ...
0
votes
1answer
54 views

cubic polynomial

I need to find closed-form solutions of the polynomial $$ y^3-\alpha_6 y^2+(\alpha_1-\alpha_2)y-(\alpha_3+\alpha_4-\alpha_5)=0, $$ where all the coefficients are real numbers. I'm worried that the ...
0
votes
2answers
41 views

Two quartics vs one single higher-order polynomial

I was reading an article called Computing the Cube Root, in which they approximate a cube root using one quartic polynomial divided by another, in the form ...
0
votes
1answer
30 views

Prove any continuous function on a 3-dim ellipsoid can be approximated by a polynomial

I'm familiar with the Weierstrass approximation theorem and some aspects of the Stone-Weierstrass theorem but I mainly only get it for closed intervals [a, b]. I am familiar with the proof that begins ...
1
vote
0answers
34 views

Example of Stone-Weierstrass Theorem on a non-interval? (3-dim compact set for example)

So I'm familiar with the Stone-Weierstrass Theorem for closed intervals [a, b] but am now looking to prove it for a more general 3-dimensional compact set. That is, if f is continuous on the set X, f ...
1
vote
2answers
22 views

Point separating function

I'm trying to prove that the set $P = \{p: [0,1] \times [0,1] \to R \; \mid \; \text{p is a polynomial}\}$ is dense in $C( [0,1] \times [0,1], R)$. I'm stuck trying to find a points separating ...
1
vote
1answer
42 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
0
votes
1answer
28 views

Zeros of polynomials and power series

Consider $k_i \in \{-1,1\}$ for every $i \in \mathbb{N}$ and consider the family of polynomials $P$ of the form $$\mathop{\sum}\limits_{i=0}^n k_it^i;$$ and the family of power series $S$ of the form ...
0
votes
1answer
35 views

Partial derivatives of polynomial in two variables

Let $k \in \mathbb N$, $a_{ij} \in \mathbb R$ for $i,j \in \mathbb N$, $i+j \le k$. A function $f:\mathbb R^2 \to \mathbb R$ $$f(x,y) = \sum_{i+j \le k} a_{ij}x^iy^j$$ is called polynomial of degree ...
2
votes
2answers
87 views

Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
0
votes
1answer
45 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
0
votes
0answers
43 views

Finding roots and studying the sign if a polynomial?

We have two polynomials $g(x):= 1+x+\cdots+x^{2m+1}$ and $f(x):= 1+x+\frac{x²}{2}+\cdots+\frac{x^n}{n}$. For the first one, we wish to find the real roots and study the sign as $x$ varies. I ...
1
vote
1answer
26 views

Show that the equation, $x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$

Show that the equation, $x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$
0
votes
0answers
26 views

Solution of a partial fraction equation

Consider the equation in the variable $\lambda$ \begin{align} \sum_{i=1}^{N}\frac{a_i}{(\lambda+b_i)^2}=1 \end{align} where $a_i$ and $b_i$ are all positive. How do you find any solution to this ...
3
votes
2answers
1k views

Proof that every polynomial of odd degree has one real root

I want to prove that every real polynomial of odd degree has at least one real root, using the intermediate value theorem. Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in ...
2
votes
1answer
125 views

Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
10
votes
4answers
289 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know we can use Weierstrass theorem but I'd like to ...
2
votes
0answers
42 views

Finding the sum of non-real roots of a polynomial.

$$x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+ \cdots a_0=0$$ Given a polynomial, how does one find the sum of non-real roots of the polynomial? I was wondering whether there is any closed form solution for ...
8
votes
2answers
261 views

How to find all polynomials P(x) such that $P(x^2-2)=P(x)^2 -2$?

I am trying the fallowing exercise : Solve $P(X^2 -2)=P(X)^2 -2$ with P a monic polynomial (non-constant) My attempt : Let P satisfying $P(X^2-2) = (P(X))^2-2$ Then $Q(X)=P(X^2-2) = (P(X))^2-2$ ...
-6
votes
2answers
122 views

sufficient and essential condition for $P(x)$ and $Q(x)$, such that $P(\sin x)= Q(\cos x)$ [closed]

What is the sufficient and essential condition for two real polynomials $P(x)$ and $Q(x)$, such that $P(\sin x)= Q(\cos x)$ for $x\in (\alpha, \beta)$, $\alpha\lt \beta$?
3
votes
4answers
86 views

Number of solutions of $P(x)=e^{ax}$ if $P$ is a polynomial

In MSE question the equation $x^2-1=2^x$ is considered, this is a generalization: Let $P_n(x)$ a polynomial of degree $n > 0$. It is well know that the equation $P_n(x)=0\;$ has at most $n$ real ...
0
votes
2answers
36 views

Determine polynomials with $n$-variables

Here is a funny problem arise from harmonic analysis: Let $E$ be a measurable subset of $\mathbb R^n$ with $m(E)>0$, where $m$ is the usual Lebesgue measure on $\mathbb R^n$. In practice, $E$ ...
3
votes
2answers
139 views

Partial derivatives, polynomial with two variables

I have problems proving the following result: Each $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $\forall a,b \in \mathbb{R} \ : \ f_a(y) := f(a,y), \ f_b(x) := f(x,b) $ are polynomials is a ...
2
votes
1answer
81 views

Proving set of polynomials of degree less than $n$ is closed

$P_n$ is a subspace of $C[0,1]$ where the norm is defined as $\|f-g\| = \sup |f-g|$ where $x$ is restricted to $[0,1]$. In addition the coefficients are reals restricted to the domain $[0,1]$. How ...
2
votes
1answer
74 views

conditions for the existence of complex roots:

find the necessary conditions under which the following polynomial will have non-real roots: $P(x)=Ax^3+Bx^2+x-D$ where $A>0$ and $D>0$. well if it has a+ib and a-ib as conjugate root then the ...
0
votes
1answer
118 views

Prove there are 3 real roots to this equation using Rolle's Theorem

I need to prove there are $3$ real solutions to $x^5 - 4x + 2 = 0$. I know $f(-2)$ is negative, $f(0)$ is positive, $f(1)$ is negative, $f(2)$ is positive so that by IVT there are at least $3$ roots. ...
1
vote
1answer
26 views

Special polynomials having atleast one root on the unit circle

I have the following problem: For each $w\in\mathbb{T},$ ($\mathbb{T}$ denotes the unit circle), consider the polynomial $P_{w,n}(z)=z^{n+1}+z^n-2w$ of degree $n+1,$ where $n\in\mathbb{N}.$ Does there ...
1
vote
1answer
27 views

Prove that $f(x)=(e^{ix}-e^{iz_0})f_1(x)$ where $f_1(x)$ is also a trigonometric polynomial

Let $f(x)=\sum_n c_ne^{inx}$ be a trigonometric polynomial. It then makes sense to define $f$ on $\mathbb{C}$ by allowing $x$ in this formula to be any complex number. Suppose $f(z_0)=0$ for some ...
0
votes
1answer
20 views

Non-Negligible function arithmetics

Following the other question: If a function is known to be non-neligible by this definition, (for example $q(x)=1/x$, is it true (provable) that $poly(x)*q(x)$ (for ...
0
votes
1answer
38 views

Negligible function arithmetics

By definition of negligible function, if $q(n)$ is a negligible function, does $poly(n)*q(n)$ is also a negligible function? How can I prove it?
1
vote
1answer
127 views

Hermite polynomials form complete system

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Show that the $\dfrac{h_k}{\|h_k\|_2}$'s form a complete orthogonal system. We can show that $h_k(x)=H_k(x)e^{-x^2/2}$, ...
5
votes
1answer
60 views

Complete orthogonal system from polynomials

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Show that the $\dfrac{h_k}{\|h_k\|_2}$'s form a complete orthogonal system. (Hint: We have $\langle Af,g\rangle=\langle ...
2
votes
3answers
121 views

angle between polynomials

let $v$ be the space of polynomials less than or equal to three and let $$\langle p,q\rangle = p(0)q(0)+p'(0)q'(0)+p(1)q(1)+p'(1)q'(1)$$ What is the angle between the polynomials $2x^3-3x^2$ ...
8
votes
3answers
126 views

sufficient condition for a polynomial to have roots in $[0,1]$

Question is to check : which of the following is sufficient condition for a polynomial $f(x)=a_0 +a_1x+a_2x^2+\dots +a_nx^n\in \mathbb{R}[x] $ to have a root in $[0,1]$. $a_0 <0$ and ...
2
votes
0answers
65 views

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$? In this paper on quantum computing (last paragraph of page 25), Dorit ...
2
votes
2answers
39 views

Prove inequality for polynomials

Let $a_0,a_1,\dotsc,a_n \in \mathbb C$ and $p(z) = a_0+a_1z+\dotso+a_nz^n$. How can one show that $\lvert p(z)\rvert \ge \lvert a_n\rvert\lvert z\rvert^n-\big\lvert\sum_{j=0}^{n-1}{a_jz^j}\big\rvert$ ...
4
votes
1answer
83 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
15
votes
2answers
349 views

Necessary and sufficient conditions for a polynomial $p$ to satisfy $\|x\|\to\infty\implies p(x)\to\infty$?

I'm looking for a necessary and sufficient conditions (I'm not even sure these exist) for a polynomial $p:\mathbb{R}^n\to\mathbb{R}$ to be "radially unbounded", that is $$\|x\|\to\infty\implies ...
3
votes
2answers
69 views

Question about polynomial $\sum_{j=1}^n j^k$

How could I prove that $ 1^k + 2^k + \cdots + n^k \in \Theta(n^{k+1}) $ or, equivalently, $$ 0 < \lim_{n\to\infty}\frac{\sum_{i=1}^n i^k}{n^{k+1}} < \infty? $$ I would appreciate a hint rather ...