Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

learn more… | top users | synonyms

1
vote
0answers
84 views

Time complexity of Gaussian elimination over polynomial ring

I have a $t \times l$-polynomial matrix $A$ over $\mathbb{F}_q[x]$. The entries of $A$ are of degree $\le m$. I want to reduce $A$ to upper-triangular form by Gaussian elimination in case of using the ...
1
vote
1answer
195 views

GCD of two polynomials in Mod 2 [duplicate]

Possible Duplicate: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ Let $p$ and $q$ be distinct primes. I wonder is the following statement always true? $$\gcd(x^p-1, x^q-1) ...
6
votes
4answers
473 views

First examples in Galois theory

I'm studying Field Theory and after studying theorems and problems about extensions, splitting fields, etc... I'm starting with the first theorems of the Galois Theory itself. In order to see if I ...
7
votes
1answer
608 views

Complex Analysis - Location of roots of a polynomial

How many roots does the polynomial $z^4 + 3z^2 + z + 1$ have in the right-half complex plane (i.e. $Re(z) \gt 0$)? I honestly can't think of how to approach the problem as it seems different from the ...
1
vote
1answer
188 views

Inequality involving roots of a third degree polynomial

Let $a,b$ be two positive numbers such that $a^3 \gt 27b$. Consider the polynomial $$ W(x)=x^3-2ax^2+a^2x-4b $$ Then we have $$ W(0)=-4b \lt 0, \ W(\frac{a}{3})=\frac{4}{27}(a^3-27b) \gt 0, \ ...
1
vote
2answers
75 views

Non linear curve

If there are set of points (16.26,1000) (8.814,2000) (6.06,3000) (4.602,4000) in a similar way if you plot these points in matlab you will get a nonlinear curve. After getting that curve, how to find ...
0
votes
1answer
80 views

Prove the polynomial is irreducible on the prime field $F_2$

How to prove that $$x^{2^n}+x+1$$ is irreducible in $F_2$ -Is this question relevant to finite field?
1
vote
1answer
2k views

How to calculate Inverse Z-Transform by long division

I am studying Feedback Control of Computing Systems. (specifically using Hellerstein's book, section 3.1.4, page 74) An inverse Z-Tranform also can be obtained by a long division. In the book there ...
2
votes
2answers
74 views

Compounding with varying principal?

The question pertains to determining average rate of return per year over $n$ years when the final amount and principal invested each year is known and it is assumed that the principal is invested at ...
1
vote
1answer
144 views

expressing product as Vandermonde determinants

Is it possible to express the product: $$ \frac{\prod_{i < j} (a_i - a_j)(b_i - b_j) }{\prod_{i,j} (a_i - b_j) }$$ as the determinant of a single matrix ? This comes from a physics paper. Should ...
-2
votes
4answers
110 views

Complete instead of Complex, Irregular instead of Imaginary

Will the terms complex and imaginary ever be replaced? At least within beginning classes? I imagine it is more of a kind of hazing into the "mathemitician's club" to allow the terms to confuse ...
1
vote
1answer
387 views

Multiple roots of a polynomial in two variables

Let $F\in\mathbb{C}[X,Y]$ be an irreducible polynomial and $n\in \mathbb{N}$, $n\ge1$, $p_i\in\mathbb{C}[X]$ for $0\le i\le n$, such that $$F(X,Y)=\sum\limits_{i=0}^{n}p_i(X)Y^{n-i}.$$ Let ...
1
vote
1answer
208 views

Factoring a polynomial with big integer coefficients and some known factors.

I have the following polynomial that I want to factor $$ \begin{align*} p(x)= &- 236364091 x^{13}- 28363690920 x^{12}- 1487737229594 x^{11}\\ &- 44880832661940 x^{10} - 860924276925225 x^9- ...
10
votes
3answers
2k views

Irreducibility of polynomial if no root (Capelli) [duplicate]

Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$. This answer to a related question ...
1
vote
1answer
114 views

polynomial in several variable whose maximum modulus on the ball is known exactly

I'm interested in polynomials in several variables $p(x_1,\ldots,x_n)$, with complex coefficients, such that the maximum modulus of $p$ on the unit complex $n$-ball $$ \max \{ |p(z_1,\ldots,z_n)| : ...
4
votes
3answers
133 views

The irreduciblity of $X^{2r}-X^{r}+1$

The question is that $e^{2\pi i/6r}$ is a root of the polynomial $X^{2r}-X^{r}+1 \in \mathbb Q[X]$ , we want to prove that $X^{2r}-X^{r}+1$ is irreducible if and only if $r$ is of the form ...
7
votes
2answers
149 views

One polynomial problem

It's known that a polynomial $f\in\mathbb{C}[x]$, whose degree is $n$, possesses integer values at each of the points: $0,1,4,\ldots,n^2$. Prove that this polynomial possesses an integer value at ...
17
votes
1answer
479 views

Irreducibility of $x^{n}+x+1$

Motivated by this problem, and KCd's comment on my answer, I am left with the following question: Question: Suppose that $n\not \equiv 2\pmod{3}$. Is $$x^n+x+1$$ irreducible over $\mathbb{Q}$? ...
3
votes
2answers
395 views

Synthetic division via the greedy strategy

I was looking at the expanded synthetic division within Wikipedia. I was stumped by how to come up with and perform the 'compactified' version of synthetic division. Does anyone know how to do it?
6
votes
2answers
414 views

Help me solve this olympiad challenge?

Given: $$p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$$ What is the sum of all arctan of all the roots of $p(x)$?
3
votes
2answers
480 views

Checking if a System of Polynomial Equations is Consistent

I'm trying to determine whether any solutions exist to a system of $(n+1)$ polynomial equations in $n$ unknowns. For example: $$ \begin{align*} xy&=-2\\ x^2-1&=0\\ y^3-3y^2+2y&=0 ...
4
votes
1answer
135 views

A “known” polynomial sequence?

Some published papers and books give the impression that if you write down any infinite sequence of polynomials that follows a simple pattern, one will find that it's named after somebody and has an ...
2
votes
1answer
604 views

Plotting $(x^2 + y^2 - 1)^3 - x^2 y^3 = 0$

I have no idea how this equation: \begin{equation} (x^2 + y^2 - 1)^3 - x^2 y^3 = 0 \end{equation} Produces this picture: Can someone provide a general explanation of plotting this function?
11
votes
1answer
1k views

Zero divisors in polynomial rings

The following is an exercise in Hungerford (Ch. III, ex. 5.6). Let $R$ be a commutative ring with identity. If $f=a_nx^n+\dots+a_0$ is a zero divisor in $R[x]$, then there exists a nonzero $b$ ...
50
votes
7answers
8k views

Prove every odd integer is the difference of two squares

I know that I should use the definition of an odd integer ($2k+1$), but that's about it. Thanks in advance!
3
votes
1answer
73 views

Controlling the coefficients of the factors of a polynomial with integer coefficients

Let $P\in {\mathbb Z}[X]$ be a polynomial, $$ P=\sum_{k=0}^{n} a_kx^k $$ Let us put $$ || P || = \max_{0 \leq k \leq n} |a_k| $$ Let $Q$ be a factor of $P$. Can we bound $||Q||$ by some function ...
0
votes
1answer
836 views

Coefficients of a cubic equation having one positive real root and two complex root with negative real part

Let $0 \lt \alpha \lt 1$ and $\beta,\gamma \gt 0$. Let $p(x) =x^{3}-\gamma x^{2}-\alpha x-\frac{\beta }{\gamma }$. Can we choose $\alpha ,\beta ,\gamma $ such that $p(x)$ has one positive real root ...
16
votes
3answers
316 views

Find all polynomials that fix $\mathbb Q$ and the irrationals

Problem: Describe all polynomials $\mathbb{R}\rightarrow\mathbb{R}$ with coefficients in $\mathbb C$ which send rational numbers to rational numbers and irrational numbers to irrational numbers.
1
vote
0answers
196 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 ...
7
votes
2answers
6k views

Minimal polynomials and characteristic polynomials

I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices. When are the minimal polynomial and characteristic polynomial the ...
2
votes
1answer
365 views

Questions about a vector space over a finite field with a bilinear symmetric form.

This is an extension of a previously asked question: Inner Product Spaces over Finite Fields. Inner product spaces in the typical undergraduate linear algebra course are stressed to be over ...
1
vote
2answers
105 views

Understanding $\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$

I'm going through a book about algorithms and I encounter this. $$\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$$ How is this equation formed? If a theorem has been applied, what theorem ...
0
votes
1answer
222 views

Generating 'easy-to-solve' polynomial equations to test function in computer science

I have a file i need to write a unit test for that includes a number of polynomial functions, and i need to provide test data to the function to calculate and compare them with an exact ...
3
votes
2answers
79 views

$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$

Given that the equation $$p(x)=a_0x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ has $n$ distinct positive roots, prove that $$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$$ I had ...
4
votes
1answer
255 views

The set of critical values of a polynomial $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is finite

I have result: measure of the set of critical values of $f$ is zero (by Sard's theorem), where $f: \mathbb{R^n} \rightarrow \mathbb{R}$ are polynomial functions. How do you show that the set of ...
4
votes
1answer
83 views

How to find $1/x^3 + 1/y^3$?

If I am given, $x + y = a$ and $xy = b$, how would I find the value of $\dfrac1{x^3} + \dfrac1{y^3}$?
0
votes
3answers
91 views

How to factorize a cubic equation?

How should I factor this polynomial: $x^3 - x^2 - 4x - 6$
2
votes
1answer
194 views

Proof of a lower bound of the norm of an arbitrary monic polynomial

In my course I have come across the following problem: The Chebyshev polynomial of degree $n$, $T_n(x)$, is defined on $[-1,1]$ by $T_n(x)=\cos n\theta$. Let $q_{n+1}(x)$ be any monic ...
3
votes
1answer
112 views

Equivalence of two binomial type equations

Given that $$A=\sum_{i=k}^{2k-1}\binom {2k-1} ix^i(1-x)^{2k-1-i}$$ and $$B=\sum_{i=k+1}^{2k}\binom {2k} i x^i(1-x)^{2k-i}+\frac{1}{2}\binom {2k} k x^k(1-x)^k$$ I would like to prove that $A=B$ ...
2
votes
0answers
330 views

Bare minimum of points in multiple polynomial regression

I have a question on multiple polynomial regression and the absolute minimum amount of points in the different terms. The minimum amount of points required for a second order polynomial would (in one ...
1
vote
1answer
73 views

Polynomial factoring issue

I am dealing with an issue for which I do not find answer on the Internet. When I factorize a polynomial, I can get this structure: $$ (x-a)(x-b)(x-c)^2 $$ But sometimes I have seen others like: $$ ...
2
votes
2answers
527 views

Maximal ideals in multivariate polynomial rings

Maximal ideals in univariate polynomial rings $R[X]$ have a nice characterization in that they all are of the form $(E)$, for some irreducible $E\in R[X]$. This allows for a systematic way to ...
2
votes
3answers
156 views

$z$ is a root of $F$ iff $\bar z$ is root of $F$

$F(x) \in R$ and $z \in C$. I need to prove that z is a root of $F$ iff $\bar z$ is root of $F$ I can't think of a way to prove that... will love some guidance.
1
vote
1answer
123 views

Property of an operator in a finite-dimensional vector space $V$ over $R$

Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$. For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$. (a) Prove that there exists $N$ such that ...
0
votes
3answers
340 views

$R = \mathbb{Z}[ i ] / (5)$ is not an integral domain? Why?

Let $R = \mathbb{Z}[ i ] / (5)$ . How should I prove that $5 = (2+i) (2-i)$ is a prime factorization in $\mathbb{Z}[i]$? Can we deduce from this that R is not an integral domain? How? I know that ...
1
vote
1answer
128 views

Is $R = Q[x] / (x^4 - 3x^2+ 6x)$ isomorphic to a direct sum of two fields?

Let $R = Q[x] / (x^4 - 3x^2+ 6x)$. How can we prove that $x^2 + 1$ is invertible in $R$? How can we prove that $R$ is isomorphic to a direct sum of two fields?
2
votes
2answers
238 views

How should I show it is a principal ideal domain?

Let $F$ be a field. Denote $R = \{ f/g \mid f, g \in F[x], g\neq 0 \}$, which is the fraction field of $F[x]$. Choose an element $a \in F$ and set $R_a = \{ f/g \in R\mid g(a) \neq 0 \}$. Show ...
2
votes
1answer
122 views

Simplifying a polynomial by a nice recursive formula

Let define a function $g(x)= (1+x^2 )/2 $ and then define again $G_i$ where $ G_1(x) = g(x) $ and $G_{n+1}(x) = g(G_{n}(x))$ . How can we approximate $G_{2n} $ and $G_{3n} $ with respect to $G_n$ ...
5
votes
1answer
240 views

Cyclotomic Polynomials and GCD

Since Cyclotomic polynomials are irreducible over $\mathbb{Q}$, $\phi_n(x)$, $\phi_m(x)$ are coprime as polynomials in $\mathbb{Z}[x]$. Working over $\mathbb{Q}$, $(\phi_n(x)$, $\phi_m(x))=(1)$. ...
13
votes
2answers
972 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 ...