This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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10
votes
0answers
77 views

If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
0
votes
2answers
495 views

Solve $x^4+3x+20=0$ by Ferrari's method

Comparing the equation $$x^4+3x+20=0$$ With the equation $$(x^2+\lambda)^2-(mx+n)^2=0$$ we get $m^2=2\lambda,$ $-2mn=3,$ $n^2=\lambda^2-20$ Now, $4m^2n^2=9\Rightarrow ...
1
vote
1answer
46 views

Factor Completely.

Again, this question if from my final practice exam. Factor Completely. $$81x^4-256y^4$$ I'm able to get this far, How do I know which of the two factors should be factored further. ...
2
votes
2answers
53 views

Rouche's Theorem application for $z^6-5z^4+3z^2-1$ in $|z|\leq 1$

Find the number of roots of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq 1$ Taking $g(z)=1$ would be the obvious choice, but it's not the right one. The next choice would be $z^6-1$ because we know the roots ...
1
vote
0answers
19 views

Factorizarion of an algebraic expression.

I came across an expression, $8x^3-4x+1$. It was further factorized as $(2x-1)(4x^2+2x-1)$,i.e. They added and subtracted $4x^2$, split $-4x$ into $-2x&-2x$, took requisite common factors. But ...
-1
votes
2answers
36 views

Does a polynomial in two variables which establishes a bijection between

Does a polynomial in two variables which establishes a bijection between the points with nonnegative integer coordinates and natural numbers exist? porve it
2
votes
2answers
471 views

Symbolic polynomial interpolation

I'm trying to create polynomials from some symbolic points to discretize derivations. This means I'm having data like $(a, \phi(a)),\ (b, \phi(b) ) $and $(c, \phi(c))$ and want to fit a second order ...
1
vote
0answers
9 views

How to compute the ifft of a vector?

In the following post Concrete polynomial implementation it is said that the final step before obtaining the product of two polynomials is to compute the ifft of a vector. How to compute the ifft of ...
2
votes
1answer
62 views

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
0
votes
0answers
23 views

Resultant of two single-variable polynomials via long division

I need to calculate the resultant of $Q=X^{10}+X^9 + \cdots + 1$ and $P= X^3+X^2+1$ by hand, and I already know it should be $23$. I'm obviously not gonna take the naive way via the coefficient ...
3
votes
0answers
56 views

Find the polynomial $p(x)$

A polynomial $p(x)$ gives a remainder of $1$ when divided by $x^{100}$ and a remainder of $2$ when divided by $(x-2)^3$. Evaluate $p(x)$. By the Remainder Theorem, $p(x)$ can be written as ...
9
votes
1answer
142 views

Examples of rings whose polynomial rings have large dimension

If $A$ is a commutative ring with unity, then a fact proved in most commutative algebra textbooks is: $$\dim A + 1\leq\dim A[X] \leq 2\dim A + 1$$ Idea of proof: each prime of $A$ in a chain can ...
1
vote
1answer
30 views

Unique generating element of all integer polynomials that have $1+\sqrt 2$ as a root.

I have to find a polynomial $p(X)$ with root $1+\sqrt2$, so that no matter with what $q(X)\in\mathbb Z[X]$ it is multiplied, it again becomes a polynomial with that root. And probably that every ...
1
vote
1answer
18 views

What does the homomorphism $\rho: \mathbb Z[X] \longrightarrow \mathbb R$, $X\longmapsto 1+\sqrt2$ describe?

Is it the set of all polynomials with integer coefficients, where one then "plugs in" $1+\sqrt2$? For example: $$a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$$ becomes ...
-1
votes
2answers
21 views

Determine a, b & c in a (possible) Quadratic equation [closed]

given f(x)=x: 1 - f(5)=2 2 - f(3)=3 how to determine the coefficients a, b & c in the polynomial equation like: ...
3
votes
2answers
29 views

Adding Polynomials with exponents. Can't get same answer as answer key.

This question is from my final practice exam. simplify and express each answer using positive exponents only. $$(2x^3y^{-2}z^0)^2+8x^{-3}y^2 $$ After working out the problem this is the answer that ...
1
vote
2answers
20 views

Laguerre's Method

Given that, polynomial $P(z) = \sum\limits_{i=0}^{n} a_i z^i$ where $a_i$ are the real coefficients and $P(z_0) = 0$. With the help of Laguerre's Method we find the rest of the complex solutions ...
3
votes
0answers
47 views

The GCD of a Univariate Integer-Valued Polynomial over a Set

Let $\mathcal{I}[X]$ denote the subring of $\mathbb{Q}[X]$ consisting of all integer-valued polynomials (i.e., $f(X)\in \mathbb{Q}[X]$ such that $f(k)\in\mathbb{Z}$ for all $k\in\mathbb{Z}$). For ...
-1
votes
0answers
16 views

if :$f(x)$,$g(x),q(x),R(x)$ Polynomial, $g(x) \neq 0$, $\deg f(x) \geq \deg g(x)$ [duplicate]

if $f(x)$,$g(x),q(x),R(x)$ Polynomial, $g(x)\neq 0$, $\deg f(x)\geq \deg g(x)$ then prove: $$f(x)=g(x) \times q(x)+R(x)$$
1
vote
1answer
16 views

Finding the square root of a ring element modulo a polynomial

Suppose I am working with polynomials in $\mathbb{Z}_5$. Let $P(x)$ be some irreducible quadratic. We know that the remainders modulo $P(x)$ will form a ring of remainders. Now suppose I wanted to ...
6
votes
1answer
83 views

How to tell if a system of polynomial equations has no real solutions

I have a system of $3n + 3$ polynomial equations in $6n$ variables, where $n$ is probably going to be less than about $5$. I can compute its Groebner basis and I see that it does not contain $\{1\}$, ...
0
votes
0answers
16 views

Question about $F(x,y)=m$

Let $F(x,y)$ be a homogeneous polynomial of degree $\ge3$ with mutually prime coefficients, then we consider the problem $$F(x,y)=m\tag1$$ such that $m$ is an integer, we set $f(x):=F(x,1)$ then ...
0
votes
2answers
38 views

Power of a polynomial in Galois field

Let $f(x) \in GF[q](X)$, where $q = p^m$ and $p$ prime. Is the following true? $$f^{p^m}(X) = f(X^{p^m}).$$ I tried to prove the assertion above and got stuck at the following: $$ \begin{align} ...
2
votes
1answer
529 views

Eigenvalues of $3\times 3$ Covariance Matrix, Geometric Interpretation

Problem Definition I would like to code an algorithm for decomposing a covariance matrix into its eigensolution (set of eigenvalues and corresponding eigenvectors. In my specific case I want to deal ...
-2
votes
1answer
22 views

Real polynomials from repunits to repunits ( Putnam 2007 A4) [closed]

Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
0
votes
1answer
68 views

Approximation of a polynomial with fractional power

I have a polynomial I need to find the roots of, the major difficulty is that this polynomial has fractional exponents. I have made an approximation and I would like to have some idea of the error I ...
-1
votes
2answers
51 views

General questions about Polynomial Rings [closed]

I'm learning about polynomial rings in my class. My instructor and book are both spectacularly unhelpful and didn't even bother to define most of the terms in my homework. So I have some general ...
3
votes
2answers
212 views

a general continued fraction satisfying $\frac{(i+\Theta\sqrt{z})^m}{(i-\Theta\sqrt{z})^m}=\frac{(ik+\sqrt{z})^{m+1}}{(ik-\sqrt{z})^{m+1}}$

Given any natural number $m\gt2$, let $z$,$k$ be complex numbers, where $i=\sqrt{-1}$ and consider the general continued fraction $$\Theta(k,z,m)=\cfrac{(m+1)}{km+\cfrac{z(0m-1)(2m+1)} ...
8
votes
1answer
63 views

Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows: $p_0 = a_0$ $p_1 = a_0x+a_1$ $p_2 = a_0x^2+a_1x+a_2$ $p_n = ...
1
vote
1answer
29 views

Polynomial problem involving divisibility, prime numbers, monotony

Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio ...
4
votes
1answer
35 views

The biggest possible degree of a polynomial given a condition

Let $P(x) \in R[x]$ be a polynomial with real coefficients such that $$(\forall n \in \mathbb N)(\exists q \in \mathbb Q)(P(q)=n)$$ What's the biggest possible value of $\deg P$? ($\deg P$ is the ...
5
votes
1answer
100 views

Polynomials with $S_n \times \mathbb{Z}_2$ symmetry

Suppose that a polynomial $p(x_1\ldots x_n, y_1\ldots y_n)$ in $2n$ variables is invariant under the following operations: 1) $p(x_1\ldots x_n, y_1\ldots y_n)=p(y_1\ldots y_n, x_1\ldots x_n)$ 2) ...
0
votes
1answer
28 views

Interesting 4th order factoring question

$$ A = \frac{(4\cdot2^4 + 1)(4\cdot4^4 + 1)(4\cdot6^4 + 1)}{(4\cdot1^4 + 1)(4\cdot3^4 + 1)(4\cdot7^4 + 1)}$$ What is the value of $ \dfrac{113A}{61}$ ? So i tried factoring this ...
0
votes
0answers
26 views

Polynomial Path [duplicate]

Let $x(t)$ and $y(t)$ be real polynomials in $t$. Show that there is always a polynomial relation $f(x,y)=0$. This question is taken from Artin, Algebra, Chapter 3 Vector Spaces. I have no idea how ...
2
votes
0answers
39 views

Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
0
votes
1answer
39 views

Irreducible Polynomial over $\mathbb{Z}[X,Y]$

I'm trying to see if the following polynomial is irreducible over $\mathbb{Z}[X,Y]$: $P(X,Y)=X^2Y^3+XY^2+XY+8$ Is there any simple algorithme to prove it ? Thanks
0
votes
3answers
31 views

find a and b using the information given

I have been presented with the following question : The polynomial $$f(x) = x^3 - 2x^2 +ax + b$$ satisfies the following : a) It is divisible (x-1) b) it leaves a remainder of -24 when divided ...
-1
votes
1answer
39 views
3
votes
1answer
41 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
1
vote
1answer
36 views
2
votes
0answers
34 views

A function that is locally a quotient of polynomials but not globally [duplicate]

Let $X =\{ x_1x_4=x_2x_3\;, (x_2,x_4) \neq (0,0)\} \subset \mathbb{C^4}$, i.e. not both of $x_2,x_4$ are zero. Define a function $\phi$ on $X$ by $\phi(x)=\left\{\begin{matrix} \frac{x_1}{x_2} ...
1
vote
1answer
34 views

Polynomial having all integral coefficients $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$

Let $a,b,$ and $c$ denote three distinct integers, and let $P_n$ a polynomial having all integral coefficients. Show that it is impossible that $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$. I started ...
3
votes
1answer
643 views

Irreducible polynomials over the reals

Everybody knows that the degree of irreducible polynomials over the reals is either one or two. Is it possible to prove it without using complex numbers? Or without using fundamental theorem of ...
0
votes
0answers
54 views

What does the irreducible polynomial over $\mathbb{Q}$ and $\mathbb{Q}(i)$ mean?

So here'as the problem Find the monic irreducible polynomial $g(x) \in \mathbb{Q}(x)$ for $i+ \sqrt{3}$ over $\mathbb{Q}(i)$ Erm...huh? Okay, so a minimal polynomial $m$ is what I need, and ...
1
vote
3answers
40 views

Is it true that a polynomial is reducible over a field only if the polynomial has a zero in the field?

I am doing some practice problems for abstract algebra and have come across this idea in a couple places, but it seems fundamentally wrong. For example, in order to prove that $f(x) = x^2 + x + 1$ is ...
1
vote
2answers
23 views

Are the “weights” inside a neural network actually “terms” for a polynomial?

This just hit me today. I am not too experienced with math or neural networks, but I am trying to find out about them in my own way so I can some day understand them well. So I was thinking about how ...
11
votes
6answers
7k views

Factorize the polynomial $x^3+y^3+z^3-3xyz$

I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
0
votes
3answers
86 views

If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$.

"If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, prove that $x^{16}>15$." The above problem came on a local question paper. I tried to solve it by factorizing and sum of G.P. , But I was unable to ...
1
vote
2answers
21 views

The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors

Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial). When revising some Linear Algebra topics, I got stuck with this ...
2
votes
1answer
343 views

Testing for irreducibility over $R=\mathbb Q[x]/(1+x^2)$

Let $R=\mathbb Q[x]/I$ where $I$ is the ideal generated by $1+x^2$. Then is $y^2 +1$ is irreducible over $R$ ? $y^2+y+1$ is irreducible over $R$ ? $y^2-y+1$ is irreducible over $R$ ? $y^3+y^2+y+1$ ...