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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1answer
15 views

Show that the set of all polynomials $f$ in $F[x]$ such that $f(a)=0$ is an ideal

let $A$ be an $n \times n$ matrix over a field $F$. Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal. I don't understand how to apply this when it comes to matrices. ...
-1
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2answers
20 views

For complex polynomials $\gcd(f,g)=1$ if and only if $f$ and $g$ have no common root [on hold]

Assuming the fundamental theorem of algebra, prove the following. If $f$ and $g$ are polynomials over the field of complex numbers, then $\gcd(f,g)=1$ if and only if $f$ and $g$ have no common root.
0
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1answer
6 views

Zariski closure of the subset of a line.

Let $K$ be any field, and $F=V(f)$ a Zariski closed subset of $K^n$ such that $F\cap L$ is infinite for a given line $L$. Why does $L\subset F$ hold ? If $n=2$ for instance, we may assume that the ...
-1
votes
0answers
43 views

Are polynomial roots special? [on hold]

Most functions have roots, and relations can also have roots (including complex roots) as well. They are where $f(x)=0$ for some $x$. (I'm considering only functions that actually will have roots.) ...
0
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0answers
18 views

How to find quadratic residues in the polynomial ring $k[t]$?

I have a question: Given a field $k$, finite or infinite, and an element $p(t)$ in the polynomial ring $k[t]$. I am searching for results of any kind about how to find quadratic residues in the ...
1
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3answers
69 views

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$.

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$. Obviously since this is a 5th degree polynomial, solving it is not going to be possible (or may be hard). ...
0
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0answers
12 views

If two monic polynomials have no common roots, are the coefficients of their product locally diffeomorphic to the product of the coefficients?

Let $P^d (t,\lambda)$ be the "generic" d-th degree monic polynomial $P^d (t,\lambda) = t^d + \sum\limits_{i=1}^d \lambda_i t^{d-i}$ with real coefficients. Let $\lambda(\xi,\eta)$ be given by the ...
0
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0answers
17 views

Prime and Maximal Ideals of $\mathbb{Z}[x]$ [duplicate]

Consider $R=\mathbb{Z}[x]$. Also let $p$ be a prime. Then we want to find all the prime and maximal ideals of $\mathbb{Z}[x]$. The prime ideals are $(0), (p), (x)$ and $(ap + bx)$. Then we see that ...
0
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2answers
20 views

how to find out the following statements are true or false?

Let $p(x)$ be an odd degree polynomial and let $q(x)=(p(x))^2+ 2p(x)-2$ a) The equation $q(x)=p(x)$ admits atleast two distinct real solutions. b) The equation $q(x)=0$ admits atleast two distinct ...
0
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1answer
45 views

how to prove whether statement is true or false?

Let $p(x)= x^n+\sum\limits_{k=0}^{n-1}a_k x^k$ and $q(x)= x^n+\sum\limits_{k=0}^{n-1}b_k x^k$ be two polynomials with real coefficients such that $x=3$ is a common root of the equations $p(x)=0$ and ...
3
votes
1answer
26 views

Why is a discrete algebraic subset of $K^n$ finite?

Let $K$ be any field. If $A$ is the zero set of a polynomial $P\in K[X]$, then $A$ is finite. This follows from the fact that $K[X]$ is Euclidian, using commutativity of $K$. Now let $A\subset K^n$ ...
0
votes
1answer
23 views

How polynomials are represented in matrix form for Univariate Polynomial. [on hold]

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
1
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1answer
44 views

A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
1
vote
1answer
45 views

Find the greatest common divisor of pairs of polynomials

I'm trying to find the greatest common divisor of $$p(x)=7x^3+6x^2-8x+4$$ and $$q(x)=x^3+x-2$$ where both $p(x),q(x)\in\mathbb{Q}[x].$ And if $d(x)=gcd(p(x),q(x)),$ I need to find two polynomials ...
0
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0answers
27 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
1
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2answers
23 views

$f\in L[x,y]$ such that $f(x,0)=0$ implies $f=y g$ with $g\in L[x,y]$?

Suppose $L$ is an infinite field (or even algebraically closed; I'm not sure if it is necessary to add that hypothesis). If we have a polynomial $f(x,y)\in L[x,y]$ and $f(x,0)\equiv 0$, does that ...
1
vote
1answer
98 views

Determine all $k$ such that $k^3+k+1$ is divisible by 11

The task is the following: Determine all $\ k\in\mathbb Z$ such that $k^3+k+1$ is divisible by 11 I assumed that "$k^3+k+1$ is divisible by 11" is saying $11|k^3+k+1$. That means I can rewrite it as ...
1
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1answer
60 views

What is my special quadratic?

Start with $f(x)=x^2+bx+c$. Then, attempt to solve for $x$ in $f(x)=x$. It is easily found that $x=\frac{1-b\pm\sqrt{(b-1)^2-4c}}{2}$. Then, start again with $f(x)=x$ and apply the function $f$ to ...
3
votes
0answers
17 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
4
votes
4answers
76 views

Real roots of the equation $\frac{18}{x^4} + \frac{1}{x^2} = 4$

I'm struggling a bit on the best method to find the real roots of the above equation. I ended up obtaining an equation of: $4x^4 - x^2 - 18 = 0$. Is this correct? From there on, how should I ...
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0answers
16 views

Does the resolvent cubic of the quartic equation always have at least 1 positive real root

I have written some code to solve for the roots of a 4th order polynomial, and in the process, I noticed that the resolvent cubic always has at least one positive real root. I can't find anywhere ...
4
votes
2answers
51 views

For which rings does a polynomial in $R$ have finitely many roots?

Which infinite rings satisfy the following Every non-zero polynomial in $R[X]$ has only finitely many roots ? Are there such rings which are not integral domains ?
0
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0answers
18 views

What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

Crossposted from MO The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even ...
1
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1answer
43 views

Checking whether a given polynomial is reducible or irreducible.

We're given the polynomial $x^{2}-2$ , and we need to prove that it's irreducible in $\mathbb Q$ but reducible in $\mathbb R$. Writing the polynomial as $(x^{2}-2) = 1.(x^{2}-2)$ , $(x^{2}-2)$ ...
1
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0answers
49 views

Solving a polynomial for cyclic roots

For any complex value of c, the following polynomial has 6 complex root values of p: $$1+c+2 c^2+c^3+p+2 c p+c^2 p+p^2+3 c p^2+3 c^2 p^2+p^3+2 c p^3+p^4+3 c p^4+p^5+p^6$$ For a general 6th order ...
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0answers
34 views

A subset of easily solved 4th degree polynomials

I've found (maybe, maybe not, but it's not on this Wikipedia or this Wikipedia) that there is a subset of easily solved quartic polynomials of the form ...
3
votes
1answer
51 views

Closed form for the sole positive root of the polynomial ${x^\alpha } + {x^{\alpha - 1}} + \cdots + {x^3} + {x^2} -p$, $p > 0$

For a study I'm making about the minimum and maximum radial values of bounded orbits in a central force system with general power law forces, I came across this special polynomial equation: ...
0
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0answers
18 views

Determining if a Polynomial is a subspace and its Basis

Hi, the question is Which of the subsets of P2 given in Exercises 1 through 5 are subspaces of P2 Find a basis for those that are subspaces. So I know that P'(1) = 1b + 2c And I know that P(2) = a ...
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1answer
47 views

Polynomial Factoring problem [on hold]

Find the value of $p$ and $q$ such that $15x^3 + 26x^2 - 11x - 6$ is a factor of $15x^4 + px^3 - 37x^2 + qx + 6$.
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1answer
44 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
0
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2answers
35 views

How to prove if $5/2 < x < (5/4)(1+\sqrt2)$, then $25/(x(2x-5)\ge 8$

if $\frac52 < x < \frac54(1+\sqrt2)$, then $\frac{25}{x(2x-5)} \ge 8$ First I unpacked the conclusion to: $$ 16w^2-40w-25 \le 0 $$ I attempted to solve by manipulating the interval (squaring, ...
0
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0answers
27 views

property of complex polynomials

I can't solve the following problem: Let $p(z) = z^n + a_{n-1}z^{n-1} + ... + a_0$ be a complex polynomial of degree $n \ge 1$. Assume that there exist $j \in \{0, 1, ... n-1\}$ such that $a_j \neq ...
12
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2answers
101 views
+300

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
7
votes
1answer
105 views
+50

Proving that one has solved chess by exhibiting the zeroes of polynomials over finite fields?

My question is based on one of Scott Aaronson blog post which states that a God-like being could convinced the villagers, to any degree of confidence, that she has solved chess by answering a few ...
5
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2answers
52 views

Roots of $x^p + x + [\alpha]_p \in \mathbb{Z}_p[x]$

Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{Z}_p[x],$$ where $p$ is prime. For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least one root? And for which $\alpha, p \in \mathbb{Z}$ ...
0
votes
1answer
53 views

Maximum of polynomial [on hold]

I was studying statics and came across this problem: Find the value $\beta$ such that $P$ has a maximum value in $R^2 - 1000^2 = P^2 + 2000P\cos(75^{\circ}+\beta)$. When $R$ is constant, the ...
2
votes
1answer
26 views
+200

Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 ...
0
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0answers
40 views

How do I derive the cubic formula? (without substitutions)

I've heard of a number of ways that people have derived a cubic formula (I've even heard of a number of different ways to show the formula itself too). What I want to know is how to derive it without ...
2
votes
2answers
58 views

Show $p(x)$ is a primitive polynomial

First the definition: Polynomial $q(x) \in \mathbb{Z}_p[x]$ of degree $n$ is called primitive, iff: $q(x) \mid x^{p^n-1}-1$ $\forall k : 1 \leq k \leq p^{n}-1$ : $q(x) \nmid x^k - 1$ ...
1
vote
3answers
59 views

For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
1
vote
2answers
41 views

Linear maps of polynomials, the bases of the space and their corresponding matrix.

Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by: $$Tp(x)=(x^2p(x))',$$ for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} ...
0
votes
1answer
39 views

Can the galois group be the symmetric group, if the discriminant is a perfect square?

Let $f\in \mathbb Z[X]$ be an irreducible polynomial. Suppose, the discriminant of $f$ is a perfect square. Can the galois group of $f$ over $\mathbb Q$ be $S_d$, where $d$ denotes the degree of ...
7
votes
1answer
69 views

Prove/Disprove : Every polynomial with prime degree and coefficients in $[-1,1]$ has galois-group $S_p$

Conjecture : Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial with degree $p$ and coefficients in the range $[-1,1]$. Then the galois group of $f$ over $\mathbb Q$ ...
0
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0answers
16 views

Check if intersection of curve and line exist and find it

I'm working on moist/dry air characteristic and as far I'm not really from a mathematic background, I'm struggling with a relatively basic problem which is finding the intersection of 2 curves. I ...
0
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0answers
16 views

Is this part of a known sequence?

while trying to express as an infinite sum the function $t^x/\Gamma(x)$ I came across some coefficients of the form $a_0=1$ $a_1=-\psi^{(0)}(1)$ $a_2=[\psi^{(0)}(1)]^2-\psi^{(1)}(1)$ ...
0
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0answers
13 views

Identifying indeterminable terms in polynomial fit

I am using SVD to fit a polynomial surface to a set of points, where the number of points may be less than, equal to, or more than the number of polynomial terms. For simplicity, let's assume points ...
0
votes
1answer
41 views

Alternate proof to number of monomials in a given degree - “more” rigorous, formal [duplicate]

Let $s$ be the number of variables and $n$ be the degree of the monomials we want to count in $R[X_1,\dots,X_s]$. Then show, that the count is $$\delta(n,s):=\binom{s-1+n}{s-1}.$$ The question ...
0
votes
0answers
9 views

Entry Expansion of Power Matrix

Suppose $A:=\{a_{i,j}\}, 1\le i,j, \le n$ is a $n\times n$ matrix with real positive entries. Now replace the constant $a_{1,1}$ with a real variable $x$. Denote by $A_x$ the resulting variable-Matrix ...
0
votes
0answers
13 views

Expected values of Hermite polynomials under Gaussian distribution

On Wikipedia there's a nice result stating that $$E[He_n(X)]=\mu^n,$$ where $He_n$ is the (probabilists') Hermite polynomial of order $n$ and $X$ is a $N(\mu, 1)$ random variable. I'm interested in ...
1
vote
2answers
24 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...