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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0
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2answers
86 views

Prove: p-mq | f(m) where 'm' is any integer

How to prove that $p-mq \mid f(m)$ where $m$ is any integer, $f(x) = A_0 + A_1 x + A_2 x^2 + ... + A_{n-1} x^{n-1} + A_n x^n$, $f(x)∈ ℤ[x]$, $p/q$ is a zero for $f(x)$ and $p$ and $q$ are coprime ...
4
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1answer
50 views

Determining the multiplicative group of a ring of polynomials

Let us say that we have the polynomial ring R[x]. Would it be possible to determine the order of the multiplicative group of R[x] modulo a polynomial f?
1
vote
1answer
54 views

Newton polynomials

Consider the family of symmetric polynomials $\sum^n_{i=1} x_i^k\in\mathbf{Z}[x_1,\ldots,x_n]$. By the fundamental theorem on symmetric polynomials there is a unique Newton poylnomial ...
0
votes
1answer
56 views

Francis Galton's surname problem

I am reading a little bit about this problem and am somewhat confused in some of the justifications provided in my readings. So the problem is that we want to find the probability of the extinction ...
6
votes
2answers
258 views

Why Rational Root Theorem only works with integers

Why does the rational root theorem only work when the polynomial has integer coefficients?
1
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1answer
68 views

Polynomial Function and Polynomials.

I've got a doubt about a ring of polynomial functions. The problem starting doing this exercise of Fraleigh (The 30). Here I had to show that $P_F$ isn't necessarily isomorphic to $F[x]$. It's easy, ...
1
vote
1answer
91 views

A Practical Guide to Splines (De Boor) - Proof of Leibniz formula

In De Boor's A Practical Guide to Splines (1978) Leibniz' formula is defined as follows (p.5): If $f = gh$, i.e. $f(x) = g(x)h(x)$ for all x, then $$ [\tau_i, ..., \tau_{i+k}]f = ...
1
vote
1answer
3k views

Show that the following polynomials form a basis for $P_2$

Show that the following polynomials form a basis for $P_2$. $$x^2+1, \ x^2-1, \ 2x-1$$ Is my approach correct? To check if the set is linearly independent I took $x^2$, $x$, $x^0$ to be $K$ ...
2
votes
1answer
56 views

Proving infinitely many primes (or none) for a given polynomial, e.g. $n^4+4$

I've recently started self-studying through Niven's Introduction to the Theory of Numbers and had questions on a few of the problems. In particular, I'm not sure how to show that $n^4+4$ is composite ...
2
votes
0answers
23 views

Suppose there exist infinite subsets $X_1, . . , X_n$ of F such that f($x_1, . . , x_n$) = 0 for all ($x_1, .. , x_n$) ∈ $X_1 × · · · × X_n$.

Let f($t_1, . . . , t_n$) be a polynomial over a field F. Suppose there exist infinite subsets $X_1, . . . , X_n$ of F such that f($x_1, . . . , x_n$) = 0 for all ($x_1, . . . , x_n$) ∈ $X_1 × · · · × ...
11
votes
2answers
281 views

Show that $A[X]/(aX+b)$ is an integral domain

Let $A$ be an integral domain, $a$ and $b \in A-\{0\}$, and let $B = A[X]/(aX+b)$. Show that, if $Aa \cap Ab=Aab$, then $B$ is an integral domain. My attempt at proof (following a hint). Denote ...
3
votes
2answers
75 views

$f,g,h$ are polynomials. Show that…

Let $f,g$ and $h$ be polynomials. Show that $\gcd(f,g,h)=\gcd(\gcd(f,g),h)$. I was thinking of signing $\gcd(f,g)=d$ and then write it by using Euclid's algorithm, but I couldn't get anything proper. ...
2
votes
3answers
229 views

Polynomial division challenge

Let $x,y,n \in \mathbb{Z} \geq 3$, Find $A,B$ such that $$x^{n-1}+x^{n-2}y+x^{n-3}y^2+\cdots+x^2y^{n-3}+xy^{n-2}+y^{n-1}= A(x^2+xy+y^2)+B$$ What is the best method to approach this?
1
vote
1answer
48 views

How is integer polynomial factorization reduced to factorization over a finite field?

I've read on Wikipedia that the problem of factoring polynomials over $\mathbb Z$ can be reduced to factoring polynomials over some finite field, but I can't find any information on how this is done. ...
0
votes
1answer
12 views

Tchebychev's polynomial and vector spaces

This polynomial is defined by: $T_n(x)=cos(narccos(x)) \forall x \in [-1,1]$ I could prove a recurrence relation: $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ But i couldn't deduce from this that Tn is a ...
2
votes
1answer
113 views

Polynomial maximization

If $x^4+ax^3+3x^2+bx+1 \ge 0$ for all real $x$ where $a,b \in R$. Find the maximum value of $(a^2+b^2)$. I tried setting up ...
2
votes
1answer
56 views

Proof of the Computability of Polynomials

In studying properties of polynomial functions I have read that they are computable. The usage of the word read implies that I cannot prove this statement, and withhold using learned for this reason. ...
1
vote
1answer
51 views

Is a polynomial that vanishes a nonempty open subset of $\mathbb{K}^n$, $\mathbb{K} \subseteq \mathbb{C}$, necessarily zero?

Let be $\mathbb K$ a subfield of $\mathbb C$ and consider $\mathbb K^n$ with the Euclidean topology. If $p \in \mathbb K[x_{1},...,x_{n}]$ vanishes on a nonempty open subset on $\mathbb K^n$, is it ...
2
votes
1answer
279 views

Iran Math Olympiad 2012 (perfect power)

Prove that if $t$ is a natural number then there exists a natural number $n > 1$ such that $(n, t) = 1$ and none of the numbers $n + t, n^2 + t, n^3 + t…$ are perfect powers. There is a solution ...
5
votes
0answers
223 views

Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
3
votes
3answers
95 views

Turn iterative function into polynomial.

So, I have an iterative function that looks something like this. $$f(x_n) = (x_n + 0.08) \cdot 0.98$$ e.g. So if $n = 2$ and $x$ started at $0$, then the equation would be equal to $(((0 + 0.8) ...
0
votes
1answer
66 views

Let $f(x)=(x^2-1)^n$. Prove that for $r=0,1, … ,n$, $f^{(r)}(x)$ is a polynomial with value $0$ at no fewer than $r$ distinct points on $(-1,1)$.

Let $f(x)=(x^2-1)^n$. Prove that for $r=0,1, ... ,n$, $f^{(r)}(x)$ is a polynomial whose value is $0$ at no fewer than $r$ distinct points on $(-1,1)$. In other words, prove that $f^{(n)}(x)$. I ...
11
votes
3answers
466 views

Determinant of Abstract Matrix

Given an $n \times n$ matrix $A$, where $x$ is any real number: $A = \left[ \begin{array}{ c c c c c c c c } 1 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & x ...
2
votes
0answers
65 views

Set of polynomials of degree less or equal than $n$ is equicontinuous (or compact) over every interval $[a,b]$ using Arzela-Ascoli theorem

Define $\Pi=\{\text{polynomials of degree }\le n \text{ over } [a,b]\}$ with fixed $n$. Norm is $\|f\|=\sup_{x\in D(f)}|f(x)| $ I am trying to proof that this set is equicontinuous using ...
0
votes
1answer
30 views

Find all solutions to the congruence relation

Let $p$ be a prime and $d$ is a divisor of $p-1$ Let $a$ be an integer that is not divisible by $p$, and suppose $a$ has order $d \pmod p$. List all solutions to $x^d -1 \equiv 0\pmod p$ My attempt ...
1
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0answers
44 views

Expression for polynomial

I wonder if it is possible to find a closed form expression for following sequence: \begin{equation*} C_1=1 \end{equation*} \begin{equation*} C_2=x^2+\frac 32 \end{equation*} \begin{equation*} ...
-1
votes
1answer
48 views

Polynomial to polynomial function on (in)finite field [closed]

Let K be a field. Prove that a transformation K[x]->(polynomial functions K->K) is injective if and only if K is an infinite field. How do I approach it? It's probably a very simple problem cause ...
0
votes
3answers
46 views

Qusetion about Zero divisors in a polynomial rings

Let $x^4-16$ be an element of the polynomial ring $E= \mathbb{Z}[x]$ and use the bar notation to denote passage to the quotient ring $\mathbb{Z}[x]/(x^4-16)$. Prove that $\bar{(x-2)}$ and ...
0
votes
1answer
376 views

General formula for iterated cumulative sum

Consider the sequence $S_0$ consisting of ones: $$ 1,1,1,1,1,1,\ldots $$ Now compute the cumulative sum of this sequence, and call the resulting sequence $S_1$: $$ 1,2,3,4,5,6,\ldots $$ Proceed ...
0
votes
1answer
90 views

Modified version of Eisenstein's irreducibility criterion

I have an assignment to extend/modify (and of course prove it) Eisenstein's criterion as follows: Let $f(x)=\sum a_ix^i\in\mathbb{Z}[x]$ with $n\ge 2$ and let $p$ be a prime such that $p\mid a_i$ for ...
1
vote
3answers
67 views

Determining polynomial from roots of another polynomial

I am working on an exercize and I know how to more bruteforcely solve it through pure algebra in its simplest form, but it's such a massive mess to demonstrate so I would like to see if there is ...
1
vote
1answer
25 views

Product of Polynomials in Several Variables?

Let $p$ and $q$ be the polynomials $\mathbb R$ given by: $$p(x)=\sum_{j=0}^m a_j x^j\quad \textrm{and}\quad q(x)=\sum_{j=0}^n b_j x^j.$$ We know that $$p(x)\cdot q(x)=\sum_{j=0}^{m+n} ...
3
votes
1answer
44 views

Algebraic number with bounded coefficients

How many algebraic numbers $z$ are there satisfying $P(z)=0$ where $P(z)$ is some polynomial with integer coefficients of degree less than or equal to $n$ such that the absolute value of every ...
1
vote
1answer
18 views

Separability of $X^{q^d}-X\in\mathbb F_q[X]$ for $d\in\mathbb N$

I know that I can verify that $f:=X^{q^d}-X\in\mathbb F_q$ and $f'$ (the formal derivative) are coprime in order to establish $f$'s separability. Is there an easier way, in particular one that does ...
-1
votes
1answer
25 views

Limits that require polynom actions?

i have encountered this example one day in the exam and i could not solve it. The tip that professor gave me was x^3-2x-4 / x-2 But yet i could not understand it, nor did i know how to start it. ...
0
votes
1answer
30 views

Transposing formula, possibly polynomial

I'm working on a game which I would like to follow behaviour of an already existing game. Unfortunately they have an odd way of calculating a players the xp(x) requirement for a level(y). When y is 1 ...
-1
votes
1answer
175 views

How to remove intersection of ideal $I$ and $J$ from union of ideal $I$ and $J$

after get the intersection of ideal $I$ and ideal $J$ how to remove this intersection from union of ideal $I$ and ideal $J$ in order to do prime decomposition how can it do in maple? actually i ...
3
votes
2answers
226 views

Product of Sums: Show that the following is a Polynomial by converting it into standard form. [duplicate]

$$\prod_{k=0}^n (1+x^{2^k})$$ The given expression simplifies to $(1+x)(1 + x^2)...(1 + x^{2^n})$ I am not able to proceed further. How do I express this in Summation form?
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2answers
19 views

If $c_{n} > 0$ then $\sum_{0}^{n}c_{k}x^{k} > 0$ for some $x \in \mathbb{R}$?

Let $n \geq 1$ be an integer and let $c_{0}, \dots, c_{n} \in \mathbb{R}$. If $c_{n} > 0,$ is there necessarily an $x \in \mathbb{R}$ such that $$\sum_{0}^{n}c_{k}x^{k} > 0?$$ I just realized ...
4
votes
1answer
85 views

Extension to complex numbers

Is there an extension to the complex numbers in which $zz^* = i$ has a solution? (The star denotes conjugation.) EDIT: I'm mathematically ignorant, but I'm guessing such an extension can't be a ...
0
votes
1answer
49 views

Finding length and width from depth using factors of a cubic equation?

So I have this application question: A pool designer is creating a pool with dimensions of length width and depth that must have specific relationships amongst their scale. Because the design ...
0
votes
1answer
79 views

The root of a monic polynomial with algebraic coefficients.

Let α be a complex number that satisfies α3 + βα2 + γα + δ = 0 β, γ, and δ satisfy cubics with rational coefficients. For example, β satisfies β3 + aβ2 + bβ + c = 0. However, it is not stated that ...
-1
votes
4answers
53 views

unique real - integer polynomial

If $ f(x) = x^{10} + 2x^9 - 2x^8 - 2x^7 + x^6 + 3x^2 + 6x + 2014 $ so can anyone here proof $f(\sqrt[2]{2} -1) = 2017$ Please do it with hands not by computer help or calculator help
2
votes
2answers
102 views

Etingof Problem 5.1, “Field embeddings”

Recall that $k(y_1, \dots, y_m)$ denotes the field of rational functions of $y_1, \dots, y_m$ over a field $k$. Let $f : k[x_1, \dots, x_n] \to k(y_1, \dots, y_m)$ be an injective $k$-algebra ...
2
votes
2answers
84 views

Solution to $x^\alpha + p x = q$?

I was wondering if there was any tricks, similar in spirit to the Vieta's substitution, that would apply the equation $$ x^\alpha + p x = q, $$ where $p,q$ and $\alpha$ are real constants. In ...
0
votes
2answers
37 views

Confused about Finite fields and polynomials

I'm asked to give a polynomial that has a root over a finite field but not a root over R. My understanding is that the finite field is contained in R (more restrictive) so how can there be a root in ...
7
votes
2answers
325 views

How to solve $y^2=3x^4+3x^2+1$ for integers.

If $x,y \in \mathbb Z$ , then find all the solutions of $$y^2=3x^4+3x^2+1$$ I was asked this question by my friend who said that he encountered this while solving another problem. I have ...
1
vote
2answers
42 views

Why are the following two statements equivalent?

I was reading some time series material and I came across this: the condition for causality is that $1 - \phi_1 z - ... - \phi_pz^p \neq 0$ for all $|z| \leq 1$ ,i.e. the zeros / roots of the ...
1
vote
1answer
148 views

Minimal polynomial is invariant under field extensions [duplicate]

I saw it's been answered before but I could not understand the answers as it delved into material that I did not study yet. My question is as follows: Suppose $p \in F[x]$ is the minimal polynomial ...
1
vote
1answer
84 views

Subring generated by $x$ is an integral domain iff it is a field iff the minimal polynomial of $x$ is irreducible

Let $R$ be a ring, $K$ a subfield of $R$, and $x \in R$. Let $F(X)$ be the minimal polynomial of $x$ over $K$. I want to prove that: $K[x]$ is a field $ \iff K[x]$ is an integral domain $\iff ...