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
21 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 ...
1
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0answers
18 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 ...
0
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2answers
23 views

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

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
10 views

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. ...
0
votes
1answer
14 views

Completion of a polynomial ring

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 ...
1
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2answers
34 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, ...
7
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1answer
75 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 ...
0
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0answers
26 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?
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0answers
22 views

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

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
25 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 ...
0
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2answers
27 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 ...
0
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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
25 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
20 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 $f^{k}(\alpha) \not= 0$ ...
10
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1answer
79 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 ...
-1
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1answer
16 views

Help with equation and explain to me? [on hold]

$5(-3x - 2) - (x - 3) = -4(4x + 5) + 13$ 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 ...
0
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2answers
48 views

Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...
-4
votes
1answer
60 views

What is the inverse function of $y=x^2 + 3x +2$? [on hold]

What is the inverse function of $f(x)=x^2 + 3x +2$? Please show your solution method and demonstrate that $f(f^{-1}(x))=x$
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2answers
40 views

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $ P(49^{1/3}+7^{1/3})=4 $

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $$ P(49^{1/3}+7^{1/3})=4 $$ (Source:NYSML) My attempt Let $$ ...
0
votes
1answer
10 views

factors in polynomial rings with field coefficients

I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following: Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if ...
0
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1answer
35 views

Computing the GCD of two polynomials

I'm trying to find the $\gcd$ of $A:= x^4 - x^3 + x^2 - x + 1$ and $B:= x^2 + 2$. Using the Euclidean algorithm, I've found $$(x^4 - x^3 + x^2 - x + 1)=(x^2 - x - 1)(x^2 + 2) + (x + 3)$$ and ...
0
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0answers
25 views

polynomial multiply polynomial to get a sequential polynomial

We hava a polynomial $P(x)=\sum_{i=0}^{N-1}\omega (i)x^i,\omega(i)\in\{0,1\}$. Now, we want to get a new polynomial $Q(x)=P(x)P(x^{-1})=\sum_{i=-N+1}^{N-1}\beta (i)x^i$ and all the coefficient ...
2
votes
1answer
38 views

For how many rational $x$ is $P(x)$ such that $54x^n+P(x)=315$?

Given an integer $n >2$,for how many different rational numbers $x$ does there exist a polynomial $P(x)$ of degree $n-1$ with $P(0)=0$,and with all integer coefficients,such that ...
1
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0answers
19 views

Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
1
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3answers
29 views

Why is $\sup_{x∈[0,1]} {|p'(x)|} ≤ A_d\sup_{x∈[0,1]}{|p(x)|}$ for all polynomials $p$ of degree at most $d$?

How can one prove that for any positive integer $d$, there is a constant $A_d < 0$ such that $$ \sup_{x∈[0,1]} {\lvert\, p'(x)\rvert} ≤ A_d\sup_{x∈[0,1]}{\lvert\, p(x)\rvert}, $$ for all ...
1
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0answers
16 views

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$?

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$, which is finite and of degree at most $d$ ? Continuity of a function ...
0
votes
0answers
36 views

$R[x]$ has a subring isomorphic to $R$ [duplicate]

$R$ is a commutative ring. We need to prove that $R[x]$ has a subring isomorphic to $R$. Let $S$ be that subring of $R[x]$ which has polynomials of even degree. Now I consider a mapping from $S$ ...
4
votes
1answer
45 views

How many solution are there to equation $f(x)=f(f(x))$ given the following function?

Shown is the graph of $y=f(x)$,a polynomial function of degree $10$ whose domain is restricted to $[1,5]$.Function $f$ is symmetric about $x=3$.Compute the number of solutions to the equation ...
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votes
2answers
17 views

Why do the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$ lie in ${R}^{d+1}$'s unit sphere?

Consider the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$. Why would it suffice to suffices to assume that this vector lies in the unit ...
13
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14answers
2k views

How to prove that $k^3+3k^2+2k$ is always divisible by $3$? [on hold]

How can I prove that the following polynomial expression is divisible by 3 for all integers $k$? $$k^3 + 3k^2 + 2k$$
1
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0answers
18 views

Find GCD of polynomials over GF(101)

Hello all I'm teaching myself cryptography, and I'm struggling with polynomial arithmetic over finite fields. I've some what been able to teach myself how to do the arithmetic over $GF(2)$, but when ...
1
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1answer
28 views

Is there a rational function $f$ satisfying $f(x) =f\left( \frac{1}{1-x} \right)$ for all $x$?

I would like to find a pair of relatively prime polynomials $p,q \in k[x]$ (where $k$ is a field) such that $$\frac{p(x)}{q(x)} = \frac{p \left( \frac{1}{1-x} \right)} {q\left( \frac{1}{1-x} \right)} ...
0
votes
1answer
54 views

Is it possible to solve the following equation without using the Rational Root Theorem?

Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem? Here's my progress: $$f(x)=x^4+2x^3+2x^2-2x-3$$ ...
1
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2answers
28 views

A Subspace of the Degree $3$ Polynomials Space for which $P(5)=0$

The problem is the following: Given the space of the polynomials $V=P(x)$ of degree $3$ prove the following: The set $U$ which is defined as the set of the elements for which $P(5)=0$ is a ...
0
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1answer
42 views

If $P(x) = (x^4+x^3-3x^2+4x-4)\cdot q(x) + (2x^3-5x^2+7x-3)$ find $P(2)$

If for the polynomial $P(x)$ is true that $$P(x) = (x^4+x^3-3x^2+4x-4)\cdot q(x) + (2x^3-5x^2+7x-3)$$ find $P(2)$ I assumed that the polynomial $(x^4+x^3-3x^2+4x-4)$ has $(x-2)$ as one of its ...
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0answers
29 views

Buchberger's Algorithm Example

I've been reading Ideals, Varieties and Algorithms and came across an example of Buchberger's algorithm being computed and I am not able to understand how they came to have the final result. The ...
0
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1answer
68 views

Do there exist polynomials $f,g$ in $\mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$.

Do there exist two polynomials $f, g \in \mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$? I know that this cannot happen in $\mathbb{R}[x]$. However, since $\mathbb{C}$ is algebraically closed, this ...
1
vote
3answers
108 views

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$

Find all pairs of nonzero integers $(a,b)$ such that $(a^2+b)(a+b^2)=(a-b)^3$ My effort Rearranging the equation I have \begin{array} \space (a^2+b)(a+b^2)-(a-b)^3 &=0 \\ ...
0
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0answers
21 views

The coordinate ring of $\varepsilon: xy-1=0$ [duplicate]

I want to show that the coordinate ring $\mathbb{R}[x,y]/\mathbb{R}[\varepsilon]$ of $\varepsilon: xy-1=0$ is not isomorphic with the polynomial ring of one variable $\mathbb{R}[x]$. To me this is ...
3
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1answer
39 views

Fibonacci sequence in the factorization of certain polynomials having a root at the Golden Ratio

I was playing around with the Golden Ratio $\Phi = \frac{1 + \sqrt 5}{2}$ on Wolfram Alpha and I noticed that if $F_n$ denotes the $n{th}$ Fibonacci number, then the polynomial $P_n(x) = x^n - F_n x - ...
5
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2answers
69 views

Let $a$ be a root of the cubic $x^3-21x+35=0$. Prove that $a^2+2a-14$ is a root of the cubic.

Let $a$ be a root of the cubic $x^3-21x+35=0$. Prove that $a^2+2a-14$ is a root of the cubic. My effort Working backwards I let $P(x)$ be a polynomial with roots $a,a^2+2a-14$ and $r$. Thus, ...
0
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1answer
25 views

problems to understand a special definition of “free graded commutative algebra” from lecture

I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., ...
0
votes
1answer
17 views

Complementary subspace of $M=(p(2x)=p(x)) ,p\in P_4$

Can anyone please help me with: Find a some base for complementary subspace of $$M=(p\in P_4 : p(2x)=p(x+1)), $$
4
votes
1answer
35 views

Problem involving polynomial and arbitrary continuous function

Let $f\in C^4[0,1]$ and $p$ a polynomial of degree $3$. Suppose: $$f(0)=p(0),\quad f'(0)=p'(0),\quad f(1)=p(1),\quad f'(1)=p'(1)$$ Show that for each $x\in [0,1]$ there exists $\xi\in [0,1]$: ...
0
votes
1answer
80 views

Algebraic element - integral domain

Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k \in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that $\overline{\overline{L}}=\overline{L}$. ...
1
vote
1answer
25 views

Supremum of integral polynomial near origin

Let $P(x,y)$ be a polynomial with integer coefficients that is constant neither in the horizontal nor vertical direction. Prove that $\sup_{-2\leq x,y\leq 2}|P(x,y)|\geq 4$. I suspect we might be able ...
3
votes
1answer
28 views

Rational Points on Fibonacci-like Sequence of Polynomials

Let $\{a_n\}$ be a sequence of polynomials in $\mathbb{Q}[x,y]$ with $a_0=0,a_1=1$, and $$a_n=xa_{n-1}+ya_{n-2}$$ The first few look like $$a_3:y+x^2$$ $$a_4:2xy+x^3=x(2y+x^2)$$ $$a_5:y^2+3x^2y+x^4$$ ...
0
votes
1answer
17 views

Help finding a second homogeneous polynomial of degree 5 that are also harmonic

Essentially I have to find 2 homogeneous polynomial of degree 5 that are also harmonic. Knowing z=(x+iy) is analytic I found my first polynomial to be f(z)=z^5 and that multiples of this would ...
0
votes
0answers
13 views

Using Newton form to find quadratic interpolating (or osculating) polynomial

Let $\ g(t)=\cos(\pi/2 \times t)$ Use Newton form to find the polynomial $\ pt(t)$ which agrees with $\ g(t) $ for each of the sequences $\ [t_0,t_1, t_2] $ The given data is $\ [0,0,2]$ I am having ...