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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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 ...
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1answer
35 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 $(x^{2}-2) = 1.(x^{2}-2)$ , $(x^{2}-2)$ ...
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20 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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26 views

A subset of easily solved quartic 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 ...
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1answer
44 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: ...
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15 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
44 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
42 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, ...
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2answers
31 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, ...
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0answers
23 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 ...
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34 views

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 ...
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2answers
51 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 $\alpha, p \in \mathbb{Z}$ ...
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1answer
52 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 ...
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1answer
20 views

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 ...
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0answers
38 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 ...
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2answers
53 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: $q(x) \mid x^{p^n-1}-1$ $\forall k : 1 \leq k \leq p^{n}-1$ : $q(x) \nmid x^k - 1$ ...
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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 + ...
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2answers
39 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} ...
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1answer
31 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 ...
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1answer
65 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 $\mathbb Q$ ...
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15 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 ...
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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 $a_0=1$ $a_1=-\psi^{(0)}(1)$ $a_2=[\psi^{(0)}(1)]^2-\psi^{(1)}(1)$ ...
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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 ...
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1answer
38 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 ...
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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 ...
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12 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 ...
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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}\}$ ...
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1answer
33 views

Basis for 4th degree polynomials such that integral of $p(x)$ from $-1$ to $1$ equals $0$

Let $U= \{ p \in \mathscr P_4\mathbb{R} \ | \int_{-1}^1 p(x)dx=0\}$. a.) Find a basis for $U$. b.) Find a subspace $W$ of $\mathscr{P_4}(\mathbb{R})$ such that $\mathscr{P_4}(\mathbb{R})= U \oplus ...
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3answers
46 views

Simplifying an equation from $a - b(c)$ to $a(b) - b(c)$.

A section in a book shows jumping from $$7-3(9-7) = 1$$ to $$4(7)-3(9) = 1$$ I can't see how this happens. What steps have been taken? Where did the four come from?
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1answer
28 views
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Can a hermitian, rational polynomial have non-zero odd and real coefficients in the numerator/denominator?

Assume that we have a rational polynomial of the form: $$\chi\left(\omega\right)=\frac{\sum_{n=0}\left(c_n+ic_n^{\dagger}\right)\omega^{n}}{\sum_{n=0}\left(d_n+id_n^{\dagger}\right)\omega^{n}}$$ ...
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24 views

Irrational roots conjugate theorem

This theorem seems pretty clear cut at first, but i have read a lot of queries about it. I have found out that if a cubic has only $1$ irrational root, then it cannot be expressed in the form $a + ...
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3answers
54 views

A non continuous linear map $A:\Bbb{R}[X]\rightarrow \Bbb{R}$ such that $A(P)=P(1).$

I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$ For the cas ...
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1answer
29 views

$f$ is divisible by a square of non-constant polynomial iff $f,f'$ are not relatively prime

Let $R$ be a commutative ring and $f=a_0+ \cdots +a_nt^n \in R[t]$. Define $f':=a_1+2a_2t+ \cdots + na_{n-1}t^{n-1}$. Show that $f$ is divisible by a square of non-constant polynomial if and only ...
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0answers
34 views

$\sqrt{1-x^2}|P(x)|\le 1$ for all $x\in [-1,1]$

Let $P(x)$ be a real polynomial with degree $n$ such that $\sqrt{1-x^2}|P(x)|\le 1$ for all $x\in [-1,1]$. Prove that $|P(x)|\le n+1$ for all $x\in [-1,1]$. This question was posted some years ...
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0answers
30 views

What is the maximum value of coefficient $f_v$ with the constraint that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
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2answers
30 views

How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...
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0answers
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A determinant that arises when proving the Alternating Sign Matrix Conjecture

Prove that $$\det\bigg(\frac{1-s^{i+j-1}}{1-t^{i+j-1}}\bigg)^n_{i,j=1}=t^{n^3/3-n^2/2+n/6}\prod_{1\leq i<j\leq n}(1-t^{j-i})^2\prod_{i,j=1}^n\frac{1-st^{j-i}}{1-t^{i+j-1}}$$ In his book, D. ...
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0answers
30 views

Completion of a polynomial ring [on hold]

Let $R$ be a commutative ring with ideal $I$. Let $J$ be the ideal of $R[x]$ generated by $I$ and $x$. What is the $J$-adic completion of $R[x]$? Is it $S[[x]]$, where $S$ is the $I$-adic ...
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2answers
39 views

Given $A$, $A^{-1}$ can be expressed with: $A^{-1}=bA+dI$

Given the matrix $A=\begin{pmatrix} -1 &3 &3 \\ 3& -1 & 3\\ 3& 3 & -1 \end{pmatrix}$ then $A$ is invertible and $A^{-1}$ can be expressed with: $A^{-1}=bA+dI, ...
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1answer
102 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
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36 views

Reduction modulo $p$ of $x^4 + 3x^3 -21x^2 -62x -40$ with a multiple root

Let us consider $$g(x) = x^4 + 3x^3 -21x^2 -62x -40 \in \mathbb{Z}[x].$$ How does one find the primes $p>0$ such that the reduction of $g(x)$ modulo $p$ has a multiple root (without taking into ...
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29 views

Pick out a polynomial such that ideal $J=q(x)R$ , where $q(x)$ is polynomial and $R$ is ring [closed]

In the ring of polynomials $R =\mathbb Z_5[x]$ with coefficients from the field $\mathbb Z_5$, consider the smallest ideal $J$ containing the polynomials, $p_1(x) = x^3 + 4x^2 + 4x + 1$ $p_2(x) = ...
2
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1answer
26 views

Degree of Rational Function

This might sound like a very trivial question but I found different answers on the web. Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What is ...
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2answers
31 views

equation to create unique value

I have a list of n objects say [ apple, orange, carrot, cherry, banana ] Now I am trying to come up with an equation which will generate an unique number for ...
1
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1answer
29 views

Binomial expansion in the form $(1+x^2)^n$

I'm used to dealing with binomial expansion in the form $(1+x)^n$. I understand that if the number is not $1$ then you have to divide the whole bracket by something which would make it $1$. However ...
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0answers
11 views

Is it possible to convert a general quintic to Brioschi form in one single transformation?

The standard method of converting a general quintic to Brioschi form $X^5-10CX^3+45C^2X-C^2=0$ proceeds in two steps which required the extraction of a square root. One first converts to the ...
0
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1answer
26 views

Let $f(x)$ be a polynomial in $x$ of degree greater than $1$ defined by $g_i(x)$ .Find the average of the roots of $g_{89}(x)$.

Let $f(x)$ be a polynomial in $x$ of degree greater than $1$.Degine $g_i(x)$ by $g_1(x)=f(x)$,and $g_{k+1}(x)=f(g_{k}(x))$.Let $r_k$ be the average of the roots of $g_k$.Determine $r_{89}$ if ...
1
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1answer
23 views

Proving Multiplicity in Polynomials with derivatives.

After learning multiplicity in polynomials we were given the task of proving that: if $ f(\alpha) = f'(\alpha) = f''(\alpha) = f'''(\alpha) =$ .... $f^{k-1}(\alpha) = 0$ and $f^{k}(\alpha) \not= 0$ ...
11
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1answer
90 views

Is there a polynomial such that $F(p)$ is always divisible by a prime greater than $p$?

Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that ...
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1answer
18 views

Help with equation and explain to me? [closed]

$5(-3x - 2) - (x - 3) = -4(4x + 5) + 13$ Solve for $x.$ Can anyone solve this equation and explain it to me? I don't understand this equation for some reason and I always keep getting different ...