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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0answers
16 views

irreducible polynomials over $\mathbb{F}_p[x_1,x_2]$

Recently I was reading about a irreducibility test for polynomials over the ring $\mathbb{F}_q[x]$. It is layed on the fact that the product of all monic irreducible polynomials whose degree divides ...
1
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0answers
13 views

Minimize the rank of a matrix with some entries known

Let $m,n$ be two positive integers, with $m\geq n$. Suppose we have $m$ sets $A_1,\ldots, A_m\subseteq [n]$, with $|A_i|=d_i$. Let $\mathbb F$ be a finite field of size $q$. Let $D$ be the set ...
4
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3answers
44 views

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root.

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root. I've found one method which is to equate $$2x^3-9x^2+12x-k=2(x-r)^2(x-c)$$ Expanding and equating coefficients I ...
11
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1answer
1k views

How can I get to Mars with a polynomial?

In order to get to Mars you must win a video game. The video game chooses $10$ points $(a_i,b_i)$ where $a_i$ and $b_i$ are single-digit integers, and places a disk with radius $1/3$ on each of ...
3
votes
1answer
28 views

Alternating polynomial

Let $p(x_1,x_2,\dots, x_n)=\prod \limits_{i<j}(x_j-x_i)$ and $\sigma$ - some permutation of a set $\{1,2,\dots,n\}$. Prove that $$p(x_{\sigma(1)},x_{\sigma(2)},\dots, ...
4
votes
3answers
32 views

Prove that $a(x+y+z) = x(a+b+c)$

If $(a^2+b^2 +c^2)(x^2+y^2 +z^2) = (ax+by+cz)^2$ Then prove that $a(x+y+z) = x(a+b+c)$ I did expansion on both sides and got: $a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+c^2y^2=2(abxy+bcyz+cazx) $ but ...
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0answers
12 views

Laurent Ideal whose Intersection with Polynomial Ring Requires More Generators

I want to find an ideal $I\subseteq \mathbb Q[x^{\pm 1}, y^{\pm 1}, z^{\pm 1}]$ which requires fewer generators than the affine ideal $I\cap \mathbb Q[x, y, z]$. I tried finding a principal ideal $I$ ...
2
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0answers
20 views

How Do I Apply the Cantor-Zassenhaus algorithm to $\mathbb{F}_2$?

Recently, I've been trying to implement the Cantor-Zassenhaus algorithm in C++ over $\mathbb{F}_2$. According to this lecture, the algorithm is basically: Input is polynomial $f\in\mathbb{F}_q$ with ...
4
votes
1answer
52 views

Polynomial divides set of points

Given a set of points in the plane with distinct $x$-coordinates, each point colored black or white. A polynomial $P(x)$ "divides" the set of points if no black point lies above $P(x)$ and no white ...
1
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1answer
31 views

Multilinear Mappings

Let $E$, $F$ complex Banach spaces and $p,q\in \mathbb{N}$ with $p+q\geq 1$. I will denote by $\mathcal{L}_a(^{p,q}E;F)$ the subspace of all $(p+q)$-linear mappings $A\in ...
4
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2answers
35 views

zeros of $p(z)=z^4+2$

I want to find all zeros of $p(z)=z^4+2$ and I'm not sure if I've done everything correctly. Can you correct this if something is wrong? $$x^4+2=0 \iff x^4=-2=2\cdot(-1)$$ $$\Rightarrow x_k= ...
0
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0answers
6 views

A special case of the boolean multivariate quadratic polynomial problem

It's well known that in the general case, the boolean MQ problem: given $(f_1, \ldots, f_n) \in \mathbb{F}_2[x_1, \ldots, x_m]$ with $\deg(f_i) = 2$, can we find a solution $\vec{y}: f_i(\vec{y}) = ...
1
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2answers
80 views

Is there a formula for the roots of a Quintic Equation?

I can get my head around this so someone explain it please. $(1)$ From Galois theory it is known there is no formula to solve a general quintic equation. But it is known a general quintic can be ...
1
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1answer
51 views

Wanted: Polynomial $P(x)$ with $P(-l(l+1))=1/(2l+1)$, for $l\in \mathbb{N}$

I'm looking for a polynomial $P(x)=a_1+a_3 x+ a_5 x^2+\dots$ (numbering of $i$ in $a_i$ is due to the application of this) with sampling points $P(-l(l+1))=\frac 1{2l+1}$, for $l=1,2,3,\dots$ ...
0
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1answer
17 views

Find unique polynomial of deg $d$ s.t $(P(0), P(\frac{1}{d}), P(\frac{2}{d}), \ldots,P(1)) = a\forall a \in \mathbb{R}^{d+1}$

Let $T:\mathcal{P}_d(\mathbb{R})$ be the set of polynomials of degree max $d$ and with coefficients in $\mathbb{R}$. We define the linear map$$ T:\mathcal{P}_d(\mathbb{R})\to \mathbb{R}^{d+1} $$ $$P ...
23
votes
1answer
293 views
+50

Prove $|P(0)|\leq 2n+1$

Let $P(x)$ be a polynomial with degree $\leq n$ and $|P(x)|\leq\frac{1}{\sqrt{x}}$ for $x\in(0,1]$. Prove that $|P(0)|\leq 2n+1$. The idea should be that if $|P(0)|$ is too large, then the polynomial ...
0
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0answers
12 views

Can this implementation of the least squares regression be adapted for more precision? [closed]

I need some help to determine if a piece of VB code can be adapted. The code is as follows: ...
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0answers
23 views

How to compute? [duplicate]

Let $F=\mathbb Q[x]/\langle x^n +1\rangle$ and $n$ is an integer. Define $|f| = \sqrt{a_0 ^2 + a_1 ^2 +\cdots + a_{n-1}^2}$, where $f = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1}$. If $f \in F$ satisfies ...
3
votes
2answers
38 views

All zeroes of monic cubic $x^3+ax^2+bx+c$ are negative reals and $a\lt3$. Range of $b+c$?

$a,b,c$ are real numbers. I have to find the range of values of $b+c$. So, I started off by assuming $\alpha , \beta , \gamma$ as the roots. This gives us $\alpha \beta \gamma = -c$ and ...
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0answers
36 views

Can a cubic equation be turned into a system of simultaneous equations in three unknowns?

Whilst trying to understand the role of symmetric polynomial s in Galois' Theory I managed to solve a quadratic equation by converting it into a system of simultaneous equations in two unknowns. I ...
0
votes
2answers
25 views

Prime ideals in polynomials over a field [closed]

Let $k$ be an algebraically closed field and $A = k[x_1,x_2,..., x_n]$. Then how can it be proven that the ideals $p_i = (x_1, x_2, ... , x_i) \subseteq A$ are prime for all $1 \leq i \leq n$ ? I know ...
0
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0answers
32 views

Question on invariant polynomials.

Consider the polynomial ring $\mathbb{C}[x]$ and the the cyclic group of order $14$, $C_{14}= \{1, \sigma,...,\sigma^{13} \}$, act on $p(x)$ by $$\sigma^i . p(x) := p( \epsilon^i x)$$ where $\epsilon$ ...
7
votes
3answers
82 views

Is there a closed form for these polynomials?

Let $P_0(x)=1, P_{-1}(x)=0$ and define via recursion $P_{n+1}(x)=xP_{n}(x)-P_{n-1}(x)$. The first few polynomials are $$ P_0(x)= 1\\ P_1(x) = x \\ P_2(x) = x^2-1 \\ P_3(x)= x^3 -2 x\\ P_4(x) = x^4 - ...
0
votes
2answers
26 views

Show that $K[X]/(P)$ is the splitting field of $P$.

Let $K$ a field and $P\in K[X]$ and irreducible polynomial. The fact that $K[X]/(P)$ is a field is fine. I want to show that it's the smallest field where that split $P$. First, let show that ...
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0answers
20 views

How can I compute a bound for inverse elements? [closed]

Let $F=\mathbb Q[x]/\langle x^n +1\rangle$ and $n$ is an integer. Define $|f| = \sqrt{a_0 ^2 + a_1 ^2 +\cdots + a_{n-1}^2}$, where $f = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1}$. If $f \in F$ satisfies ...
1
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0answers
46 views

Prove that $|p(x)| \ge |a_o|\cdot|x^n|\cdot [1-\frac{1}{R}(|\frac{a_1}{a_o}|+|\frac{a_2}{a_o}|+…+|\frac{a_n}{a_o}|)] $

Let $n$ be a natural number, and let $p(x)=a_ox^n+a_1x^{n-1}+...+a_n$ be a polynomial with real coefficients and $a_o \ne 0$. Assume $|x|\gt R$, where $R$ is a fixed positive real number. Prove that ...
0
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1answer
17 views

Proof for Theorem of Upper and Lower Bounds On Zeroes of Polynomials

I'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof: Let $f(x)$ be a polynomial with real coefficients and a positive leading coefficient. ...
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0answers
14 views

Nilpotent Matrix Sign Patterns given by Existence of Nonlinear Multivariable Polynomial Solution

I am currently doing a little exploring in sign patterns in nilpotent matrices, and am trying to determine whether or not an ambiguous sign pattern has a solution (i.e permits a nilpotent matrix). ...
2
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3answers
53 views

Factorize $2a^3 - b^3 - c^3$

I need to factorize the expression $2a^3 - b^3 - c^3$. I see that one zero is achieved when $a=b=c$, but I can't find the factor(s).
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35 views

How do I see if $g$ is a polynomial or not??

Let $u$ be a real valued harmonic function on $\mathbb{C}$. Let $g: \mathbb{R^2} \to \mathbb{R}$ be defined by: $$g(x,y)=\int_0^{2\pi}u(e^{i\theta}(x+iy))\sin \theta \,d\theta$$ Which of the ...
0
votes
1answer
36 views

If $2^{x + 1} < y$, then what is the largest polynomial in $x$ that cannot be an upper bound for $y$?

Update: I have posted a follow-up question here. The title says it all. If $2^{x + 1} < y$, then what is the largest polynomial in $x$ (of maximum possible degree) that cannot be an upper ...
0
votes
1answer
54 views

Show that $f(0)=-2$ with $f\left(\frac{u^2+u-2}{2}\right)=u$ [closed]

Let $f(x)=ax^2+bx+c$ ,where $a,b,c\in \mathbb Q$. If $f\left(\frac{u^2+u-2}{2}\right)=u$, where $u^3-3u+10=0,u\in \mathbb R$, show that $f(0)=-2$
1
vote
1answer
27 views

Proving P(x) > 0 given a condition.

$P(x)$ is a polynomial function such that, $P(1) = 0, P′(x) > P(x), ∀ x > 1. $ Prove that $P(x) > 0, ∀ x > 1.$ I was trying to do by taking the P(x) in the denominator and then ...
2
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1answer
31 views

Non-linear optimization programming

How many methods do we have for non-linear optimization problems, which the target function is linear but constrains are polynomial shape? Are there methods which can solve most of them? Or what ...
2
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2answers
35 views

Polynomial with no integer roots

This is an excercise given to a kid I am tutoring, as part of a set of problems regarding polynomials. He is currently at the last class before graduation year. Let $p$ be a polynomial in $ℤ[x]$ such ...
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1answer
36 views

Finding the general Taylor polynomial formula for $f(x)=\log\left(\frac{1+x}{1-x}\right)$

I am trying to find the general form of the Taylor polynomial for $f(x)=\log\left(\frac{1+x}{1-x}\right)$. The $log$ is of base $e$ and I have rewritten the original formula as: $\log(1+x)-\log(1-x)$ ...
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3answers
59 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
0
votes
2answers
19 views

Polynomial with bounded coefficients and real root

A polynomial with degree $2n$ has all coefficients in the range $[100,101]$ and has a real root. What is the minimum possible $n$? Degree $0$ is clearly not possible. For degree $2$, the discriminant ...
2
votes
0answers
29 views

If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?

If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field? I am trying to prove that the ring of fraction $L(\alpha)$ is equal to ...
9
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4answers
482 views

How can I guarantee the unique positive root of this polynomial?

How can I guarantee the unique positive root of this polynomial? I have two polynomial, $$ x^{n+1} + x^n - 1 =0 $$ and $$ x^{n+1} - x^n - 1 =0 $$ respectively, where $n\in\mathbb{N}$. I have ...
0
votes
1answer
9 views

Writing the function to maximize volume or a cylinder

A rectangular piece of paper is curled into a cylinder with two open circles on each side. The perimeter of the piece of paper is 124 inches. What is a function that could be written to find the ...
0
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0answers
23 views

Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
2
votes
4answers
47 views

Elementary symmetric polynomial task with three variables

Can anyone help me to wite this as sum or product of elementary symmetric polynomial. $$\frac xy+\frac yx +\frac xz + \frac zx +\frac yz + \frac zy =7$$ I tried to set under one fraction, but I ...
4
votes
1answer
98 views

root pattern of second degree polynomial

I'm considering the following 2nd degree polynomial for the case where the roots are complex conjugate. $ P(z) = z^2 + (f^2 + f q -2)z + (1 - f q) = (z - z_1) (z - z^*_1) $ where f and q are real ...
1
vote
1answer
39 views

Finding general form of Taylor polynomial for function $f(x)=e^{x}\sin(x)$

I am trying to find the general form the Taylor polynomial of the function $f(x)=e^{x}\sin(x)$. I have calculated the derivatives up to $5$: $$\begin{align} f^{(1)}(x)&=e^{x}\cos(x) + e^{x} ...
1
vote
2answers
31 views

Type of polynomial where leading coefficient is to the power of $6$ [duplicate]

I need to identify the type of polynomial that a polynomial is based on the power of the leading coefficient. (Example $x^2$ = quadratic, $x^3$ = cubic, $x^4$ quartic). In this case, it is $x^6$. What ...
0
votes
0answers
25 views

Is there an “easier” way to find the factors of a polynomial than using Ruffini's method?

I am a first year Mathematics student, and sometimes I have a hard time with Ruffini's method for polynomials, specially in the field of Rationals
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0answers
12 views

Concave property on elementary symmetric polynomials

Let ${\sigma _k}$ be the k-th elementary symmetric polynomial, namely ${\sigma _k}({x_1},...,{x_n}) = \sum\limits_{1 \leqslant {i_1} < ... < {i_k} \leqslant n} {{x_{{i_1}}}...{x_{{i_k}}}} $ ...
1
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0answers
25 views

Clarifications regarding Lagrange resolvent

I'm trying to understand the technique used by Lagrange to solve cubic and quartic equations. I have read that the Lagrange resolvent for the cubic is $$ x_1+\omega x_2+ \omega^2 x_3 $$ where ...
0
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0answers
17 views

Root of interpolated polynomial when y-coordinates are permuted

Hypothesis: All values and polynomials are defined over a field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit) Suppose we have $n$ pairs of $(x_i,y_i)$. As we all know, given the ...