Tagged Questions
2
votes
1answer
20 views
For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
4
votes
5answers
156 views
Checking whether a polynomial of high degree is bijective or not.
Let $P(x)$ be a polynomial of degree $101$. Then $x\mapsto P(x)$ cannot be a one-one onto mapping, i.e., bijective function from $\Bbb{R}$ to $\Bbb{R}$. True or false?
I think is when we take ...
6
votes
6answers
181 views
Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$
Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function.
Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
1
vote
1answer
32 views
Finding nth degree polynomial functions
I need to find an nth degree polynomial function that has real coefficients using the following conditions:
n=3; 3 and 4i are zeros; f(2)=40
I have no idea what I'm doing on this one. It's been too ...
0
votes
1answer
39 views
Finding a function which fits this data?
I need to find a polynomial (or other continuous elementary function) on the interval [70, 180] such that it passes through the points (70, 0) (this is a relative min), (105, 17) (this is a relative ...
1
vote
6answers
126 views
Proof that if $\forall a f(a) = g(a)$ then $f=g$
How do we prove formally that if:
$\forall a f(a) = g(a)$
$=>$
$f=g$
when
$f,g \in \mathbb F[x]$
1
vote
4answers
78 views
Finding the value of a polynomial at zero
Given a polynomial $p_n(x)\,$ with $\, n \geq 0 \,$ such that $\, p_0(x)=1 \;,\; p_1(x)=x\;$ and $\; p_n(x)=xp_{n-1}(x)-p_{n-2}(x)\;$ , how can I find $\;p_{10}(0)\; $ ?
1
vote
1answer
181 views
factor-square property (FSP) of polynomials
The Factor Square Property (FSP) is the divisibility of the polynomial $f(x^2)$ by $f(x)$.
Is $x^2+x+1$ the only FSP irreducible polynomial of degree $2$ ?
Are there other linear polynomial besides ...
3
votes
3answers
76 views
Periodic polynomial?
I was thinking if it was possible to create a polynomial that would be periodic all over the reals, since polynomials can be periodic on an interval. I then I found out the following function:
...
1
vote
1answer
58 views
Recurrence Equation with Polynomial Coefficients
As inspired by this question on the problem site Brilliant,
Let $F_n(a,b,c)=a(b-c)^n+b(c-a)^n+c(a-b)^n$
Is it possible to obtain $F_n$ in terms of $F_3, F_2$?
My attempt at a solution is as ...
20
votes
2answers
380 views
How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?
How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible.
I proceeded by using the ...
0
votes
3answers
216 views
Sum of the values of $a$ for which $g^2=f^3-12f^2+45f-a$ has non-constant solutions $f,g\in\mathbb{R}[x]$
Find the sum of the values of the parameter $a$ such that for some non-constant polynomials with real coefficients $f(x)$ and $g(x)$,
$$g(x)^2 =f(x)^3−12 f(x)^2 + 45 f(x) − a$$
My understanding:
...
1
vote
0answers
101 views
How to parametrize a function such that it approaches $f(0)=0$ and $f(1)=1$ with different speed
I need a function (polynome) that values $0$ at $0$ and $1$ at $1$ and has these values as local maxima and minima. So far so easy the straight solution is:
$$f(x) = -x^4+2x^2$$
Now I want to ...
12
votes
2answers
623 views
Inverse function of $y=\frac{\ln(x+1)}{\ln x}$
I've been wondering for a while if it's possible to find the inverse function of $y=\frac{\ln(x+1)}{\ln x}$ over the reals. This is the same as finding the positive real root of $x^y-x-1$. I realize ...
-1
votes
1answer
77 views
simple arithmetic problem but..
i am having problem in this proof. i need to find the certain coefficients of this statement on the right side.
given:
$P: \mathbb{C} \Longrightarrow \mathbb{C}, \quad P(x) := 12 − 7x + x^2$
$Q : ...
1
vote
2answers
90 views
Finding the degree and coefficients of the Polynomial
A polynomial is denoted by $f(x)$. The coefficients of the polynomial are positive integers.
$$f(1) =17$$
$$f(20)=421350$$
Could you tell if such a polynomial is possible? If ye, find the degree of ...
0
votes
2answers
64 views
Power functions and parabola issue
With the function f(x)=x^2 we get a graph like so...
The rule for power functions, that I've been told, is the larger the power gets, the closer the line will touch the x-axis.
Example for ...
2
votes
2answers
368 views
Construct a polynomial function with the given graph
Where does one begin? I can see that the zeros are -5, -3, 0, and 4? Is that correct so far?
1
vote
3answers
43 views
Computational complexity proof
I would like to know how to prove the following:
$2^n \in O(n!)$
I know that I have to show that for a constant C, we have $2^n \leq C*n!$
Right?
1
vote
1answer
142 views
Using Lagrange's Interpolation Formula to show that boolean functions over finite fields are polynomials
Let $F_2$ be the set of all the functions from the finite field $GF(2^n)$ of $2^n$ elements to $GF(2)$. I am reading a textbook that proves that the elements of $F_2$ can be represented by ...
1
vote
2answers
177 views
Bell-shaped polynomial over a limited domain
The function $f(x) = e^{-x^2}$ has a bell-shaped peak at $x=0$ and then approaches an asymptote at $y=0$. I need to achieve a similar result, but with a polynomial function.
I can use a series ...
0
votes
2answers
67 views
Help to show if the function is decreasing for large $l$
I would like to see if
$$
b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.}
$$
is decreasing when $l$ is large enough say around $10^6$. I dont need any ...
3
votes
2answers
204 views
Solving a non-integer polynomial where the exponent is greater than one
I'm trying to solve an equation of the form:
$ax + bx^{1+c} + d = 0$, where $0 < c < 1$, and the reciprocal of $c$ is not necessarily an integer either.
Mathematica protests to me that it is ...
1
vote
1answer
108 views
Intersection of a line with a curve
I have the following question:
Given a line $y=\theta(1-x)$ where $0<x<1$, $0<y<1$ and $0<\theta<1$, I have a collection of curves
$$
y^K=1-(1-x)^K
$$
parametrized by an ...
2
votes
0answers
208 views
$L_2$-norm representation of the function
Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$(see for reference ...
1
vote
1answer
80 views
Greater than zero?
I need to show that
$$\sum_{i=k^*}^K\binom{K}{i}a^{i-1}(1-a)^{K-i-1}(i-aK)>0$$
given $K\geq k^*$, $0<a<1$ and $K$, $k^*\in\mathbb{Z^+/1}$.
I did some computer simulation and saw that it ...
0
votes
1answer
140 views
a multivariate quadratic function
Assume a vector-valued function, for example ${\bf f}=(f_1, f_2)$, where
$$f_1(x,y)= x^2+3xy$$
$$f_2(x,y)= 2xy+y^2$$
(here f is column vector, x, y are variables)
Assume that each $f_i$ is a ...
1
vote
1answer
97 views
Algorithms for deciding whether a function over a finite ring is polynomial or not?
Let $R$ be a finite ring, and $f$ be a function from $R$ to $R.$
Suppose I want to know whether $f$ can be represented as a polynomial or not? Are there any good algorithms for finding this out?
6
votes
1answer
377 views
Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input
$\alpha _n ^n-1=0$
$\alpha _n=e^{2 \pi i/n}$
$$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$
I have read in Jim Brown's paper on page 5 that ...
1
vote
2answers
225 views
How do I factor a polynomial function with a degree higher than 2 without guessing numbers of $\frac{p}{q}$?
I have an equation $f(x)=x^4+4x^3+2x^22-x+6$. In the past I was taught to factor it by getting the zeros by getting $p/q$, and start guessing zeros, and plugging them into the function. Once I got one ...
0
votes
0answers
109 views
How to find similar polynomials to satisfy certain boundary conditions on derivatives
Good afternoon mathematics members. I have one problem with polynomial functions.
For these functions, I need to find similar like in attachment. Derivative in start and end point and boundary ...
7
votes
2answers
370 views
What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?
Suppose $k$ is a field, and $k(t)$ is the rational function field. If $f(t)=P(t)/Q(t)$ for some polynomials $P(t)$ and $Q(t)\neq 0$, then the map $t\mapsto t+1$ sends $f(t)$ to $f(t+1)$.
So the ...
2
votes
2answers
12k views
Types of polynomial functions. Quadratic, cubic, quartic, quintic, …,?
I would very much like to have a complete list of the types of polynomial functions. I know that theres:
...
0
votes
1answer
261 views
Isolate a variable in a polynomial function
How would I go about isolating $y$ in this function? I'm going crazy right now because I can't figure this out.
The purpose of this is to allow me to derive $f(x)$ afterwards.
$$ x = \frac{y^2}{4} + ...
4
votes
3answers
344 views
Question about a recursively defined function
Problem. Let $(f_n)_{n=1}^\infty$ be a sequence of functions $f_n\colon [-1,\infty)^n\to\mathbb{R}$ that are recursively defined in the following way:
$$f_1(x_1)=1+x_1,$$
$$f_n(x_1,\ldots,x_n) = ...
4
votes
2answers
204 views
Name of property describing the number of times a function changes concavity?
For example, $f(x)=\sin x$ changes concavity an infinite number of times, $f(x)=x^3-x$ has two regions of concavity (changing concavity once), and $f(x)=x$ changes $0$ times.
Is there a name for ...
5
votes
2answers
302 views
What is an algebraic function?
I am doing first year university calculus, and we are learning about the different kinds of functions. According to wikipedia, an algebraic function ...
1
vote
0answers
96 views
Efficient way to recompute weights when shifting range of Legendre polynomial bases
I am storing a 2D (Cartesian) density function as a 2D patch with known X/Y limits and a set of 11 coefficients of the third order 2D Legendre polynomial basis functions over that patch. This works ...
0
votes
2answers
412 views
Simple asymptotic function
(I have seen this question but it is too complicated for my needs, and my math skills are not good enough to convert the answer.)
I am writing a game and I need a way to increase the armor of the ...
3
votes
5answers
324 views
How do I come up with a function to count a pyramid of apples?
My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves ...
4
votes
4answers
452 views
Is it possible to convert a polynomial into a recurrence relation? If so, how?
I have been trying to do this for quite a while, but generally speaking the partially relevant information I could find on the internet only dealt with the question: "How does on convert a recurrence ...
