Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

learn more… | top users | synonyms

-1
votes
0answers
24 views

Left remainder when dividing by $x-b$

Give a polynomial $p(x) = a_0 + a_1 x + ... a_n x^n \in \mathcal R[x]$ ($\mathcal R $ is any ring with unity), the book says when dividing $p(x)$ by $x-b \quad (b\in \mathcal R)$, the left remainder ...
2
votes
1answer
25 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
2
votes
0answers
42 views

Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
3
votes
4answers
43 views

Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain

Here is an exercise(p.129, ex.1.15) from Algebra: Chapter 0 by P.Aluffi. Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. The implication part makes no problems, ...
2
votes
2answers
31 views

Determine polynomial whose roots are a linear combination of roots of another polynomial

Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $p(x)=x^3+5x^2+7x+11$. Find a polynomial whose roots are $\frac{\alpha_1+\alpha_2}{2}, \frac{\alpha_2+\alpha_3}{2}, ...
0
votes
0answers
10 views

Efficient ways to find a single root of a multivariate polynomial system to arbitrary precision

I am looking for a practical and efficient way to compute, to arbitrary precision, a single root of a multivariate polynomial system (over $\mathbb{Q}$). It seems like the fancy methods compute all ...
1
vote
0answers
22 views

Showing polynomials as products of roots

How do I show rigorously that any polynomial $a_nx^n+a_{n-1}x^{n-1}+...a_1x+a_0$ can be written as $a_n(x-b_1)(x-b_2)...(x-b_n)$ for real $a_i$ and real or complex $b_i$
2
votes
1answer
115 views

Polynomial tending to infinity

Take any polynomial $(x-a_1)(x-a_2)\ldots(x-a_n)$ with roots $a_1, a_2,\ldots,a_n$ where we order them so that $a_{i+1}>a_i$ is increasing so $a_n$ is the biggest root. It doesn't matter whether ...
1
vote
0answers
62 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
0
votes
0answers
16 views

Maximum degree of a polynomial [duplicate]

What is the maximum degree of a polynomial of the form $\sum_{i=0}^n a_i x^{n-i}$ with $a_i = \pm 1$ for $0 \leq i \leq n, 1 \leq n$, such that all the zeros are real? I have no idea where to start.
3
votes
4answers
98 views

Find a Polynomial in $x-\frac1x$

Given that $x^n - (1/x^n)$ is expressible as a polynomial in $x - (1/x)$ with real coefficients only if $n$ is an odd positive integer, find $P(z)$ so that $P(x-(1/x)) = x^5 - (1/x)^5.$ To start, I ...
1
vote
0answers
13 views

Under what condition given $(x_1, y_1\cdot r_1),…,(x_n, y_n\cdot r_n)$ we can interpolate polynomial $T$ that has specific random root?

We know given $(x_1, y_1),...,(x_n, y_n)$ we can interpolate a polynomial $P$ of of degree at most $n-1$. Let us define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is at most $n-1$, ...
0
votes
0answers
24 views

Taylor expansion for the roots of real polynomials

Consider a (real) polynomial $\mathcal{P}$ in the variable $x$ whose coefficients are themselves polynomials in the parameter $\lambda$. I am searching a taylor expansion in $\lambda$ for the roots ...
0
votes
0answers
15 views

Connected components of pseudospectra

In this Article, page 5 Theorem 2.3 ,what is connected components of pseudospectra of matrix polynomial?
-2
votes
0answers
19 views

What is connected components of pseudospectra of matrix polynomial? . [on hold]

What is connected components of pseudospectra of matrix polynomial? Please see this link
0
votes
1answer
49 views

How many distinct roots $ax^5+bx^3+cx+d$ has

$a,b,c>0$ How many distinct roots $ax^5+bx^3+cx+d=0$ has? question doesnt clarify which kind of root it has. and I dont understand why the question didnt say 'may has' . because by ...
2
votes
3answers
140 views

Is there a name for this type of expression?

Forgive me if this seems like a silly question. I know that the following expression is an example of a polynomial: $a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ but I am wondering if there is a ...
1
vote
2answers
29 views

Dickson's Lemma (proof of Prop. 2.23 in Hasset's Intro to Alg Geom)

I'm studying Hasset's book by myself but I had no previous formal algebra training. To prove Dickson's lemma (prop. 2.23, p. 19) he defines the auxiliary monomial ideals $$J_m=\left<x^\alpha \in ...
5
votes
1answer
39 views

Why do polynomial regressions have larger variance at the end?

In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ...
0
votes
0answers
54 views

Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is fixed $n-1$, $\beta$ is chosen uniformly at random from the field of $p$ elements. We evaluate $P$ at some $x_i$ values. So we get ...
1
vote
2answers
58 views

Can Two Different Polynomials Agree on an open interval? [duplicate]

Question: For a high degree polynomial $P_1$ , can we have another polynomial $P_2$ that is a part of $P_1$ (or they agree on open interval)? TBN: This question is partially answered in ...
0
votes
1answer
59 views

Overlapping Polynomials

This question is related to this:Interpolating Polynomial & It's Root We have $P_3=P_2\cdot P_1$,for three non-zero polynomials. The degree of each polynomial is at least 1. Question: Does ...
4
votes
0answers
31 views

Irreducibility of $p(x)$ implies that of $p(x+c)$ only when taken over a field?

$R$ is a ring and $R[x]$ is the polynomial ring over $R$ . $c$ is any fixed element of $R$ . Then the map $f(x)\mapsto f(x+c)$ is an isomorphism from $R[x]$ to itself. Now ...
-1
votes
1answer
17 views

Random Permutation Polynomial With Fixed Inputs

Assume we pick uniformly random a permutation polynomial, $T$, of degree one. we define all polynomials over $\mathbb{Z}_P$. We have fixed inputs $x_i$ (e.g. $x_i \in [1,100]$) My Question: Is ...
0
votes
0answers
27 views

Why doesn't Horner's method work with the following cubic equation?

I'm trying to factor $$2x^3 - 4x^2 + 2x$$ I use the Horner's method │ 2 4 2 │ 0 --------------- │ 0 0 │ 0 --------------- 0│ 2 4 2 │ 0 and I obtain ...
0
votes
1answer
33 views

Irreducibility of a polynomial

Given that $\mathbb F$ is a field and $\mathbb F[x]$ is the polynomial ring over $\mathbb F$. $\ \ $If the polynomial $a_{0}+a_{1}x+a_{2}x^{2}+......a_{n}x^{n}$ is irreducible over ...
0
votes
0answers
11 views

Probability That a Polynomial has Specific Root when we use Permutation Polynomial

To some extent similar question was asked here: Polynomial Interpolation and Security Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-2$, ...
1
vote
0answers
53 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
0
votes
4answers
43 views

Prove by definition that $(x,2)\subset\mathbb Z[x]$ is a maximal ideal

When the polynomial ring $\mathbb{Z}[x]$ is quotiented by the ideal $(2,x)$ we get a field as $\mathbb{Z}[x]/(x,2)\cong\mathbb{Z}/(2)\cong\mathbb{Z}_{2}$ which is a field. But I ...
0
votes
0answers
34 views

Properties of Coefficients of Order Polynomials

I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain ...
0
votes
0answers
19 views

Probability that a Polynomial Has Specific Root When $y_i$'s are Not Random.

Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-1$, but degree of $P_2$ can be at most $n-1$. $P_1$ has root $\beta$, where $\beta \leftarrow ...
1
vote
2answers
28 views

Show that $D$ is surjective, but not injective.

Question: Let $P$ denote the set of polynomials $p(x) = \sum_{i=0}^n a_{i}x^i$ for $n=0, 1, 2, ....$ and real numbers $a_{i}$. Let $D:P \to P$ be the differential operator defined by $(Dp)(x) = ...
1
vote
0answers
19 views

FFT multiplication

I'm currently implementing a specific polynomial multiplication algorithm for a project. The current goal is to implement chapter 2 of Daniel Bernstein's paper ...
1
vote
0answers
17 views

How to obtain a specified set of coefficients of a multivariate polynomial

Let $x_i \in \mathbb{C}$ and $b_{ij} \in \mathbb{R}$ for all $i,j$ between $1$ and $n$. I have the following polynomial and the corresponding expansion (correct me if I am wrong): $\prod_{i=1}^n (1+ ...
0
votes
2answers
49 views

Find $\prod\limits=(\alpha_1+1)(\alpha_2+1)…(\alpha_n+1)$ where $\alpha_i$ are complex roots of a complex polynomial

The complex roots of a complex polynomial $P_n(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0$ are $\alpha_i$, $i=1,2,...,n$. Calculate the product $(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_n+1)$ By the ...
0
votes
1answer
26 views

Given that $f(x,y,x)$ is a factor of $g(x,y,z)$:

Suppose $f(x,y,z)=(x-z)^2+(x-y)^2+(z-y)^2$ and $g(x,y,z)=(x-z)^n+(x-y)^n+(z-y)^n$ Then prove that if $f(x,y,z)$ is a factor of $g(x,y,z)$, that $n$ is not divisible by 3. Please no solutions. I'm ...
1
vote
1answer
39 views

gcd of $x$ and $2$ in $Z[x]$

In $Z[x]$, $x$ and $2$ has gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2.f(x)+x.g(x)$ we are supposed to arrive at a ...
2
votes
3answers
46 views

Fastest way to perform this multiplication expansion?

Consider a product chain: $$(a_1 + x)(a_2 + x)(a_3 + x)\cdots(a_n + x)$$ Where $x$ is an unknown variable and all $a_i$ terms are known positive integers. Is there an efficient way to expand this?
3
votes
2answers
49 views

Polynomial with real roots

Consider the polynomial: $$f=X^4+4X^3+6X^2+aX+b$$ We know that $f$ has four real roots. Let $x_1,x_2,x_3,x_4$ be the roots of this polynomial. How can one compute ...
1
vote
1answer
27 views

Why mentioning monic is important for gcd and lcm?

Let $F$ is a field and $F[x]$ be the polynomial ring over $F$. Now in the definition of the gcd or lcm of any two polynomials $g(x)$ and $f(x)$ it is mentioned that the ...
0
votes
1answer
22 views

Rotated parabola 2d vertex

I'm implementing an application where I need to get the vertex of a parabola, the parabola might be tilted; so it can have an angle with the x-axis not necessarily vertical or horizontal. Can I get ...
0
votes
0answers
23 views

Asymptotic running time for multiplying multivariate polynomials using Schönhage/Strassen

Question: I would like to ask the community where my following suggestion for an asymptotic bound for the running time of multiplying two multivariate polynomials using theorem $8.23 $ recursively ...
2
votes
1answer
56 views

Interpolating a Polynomial with a Subset of Interpolation Points

Consider we has a polynomial $P=(x-\beta)g(x)$, where $\beta \leftarrow \mathbb{Z}_p$, $p$ is a large prime, and $g(x)$ is a non-zero polynomial. Here degree of $P$ is fixed $n$. We evaluate $P$ at ...
2
votes
1answer
29 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
2
votes
3answers
100 views

How to solve equation to the third power

I have the information that: $$ x^3 − x^2 −1 =0 $$ Has a "positive real root" of: $x \approx 1.4655\ldots$ My questions are, please: 1) What is a "positive real root". 2) How one gets from the ...
5
votes
4answers
101 views

How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials?

Let $x,y$ be 2 coprime integers. I assume the following polynomial:$$f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$$ is not irreducible. So there must be at least 2 other polynomials of degree $\leq 4$ such that: ...
0
votes
2answers
40 views

How to find the value of $(a+b+c)(a+b+d)(a+c+d)(b+c+d)$ from the following equation?

I have a question about polynomial. Given a polynomial: $$x^4-7x^3+3x^2-21x+1=0$$ Given too that the roots of this polynomial are $a, b, c,$ and $d$. Find the value of ...
0
votes
1answer
54 views

A particular polynomial - 2

Is there a homogeoneous polynomial in $\Bbb Z[W,X,Y,X]$ that contains only coefficients from $W^4,X^4,Y^4,Z^4,W^2X^2,W^2Y^2,X^2Z^2,Y^2Z^2,WXYZ$ that factorizes into unequal quadratic forms? What is a ...
-2
votes
2answers
53 views

A particular polynomial

Is there a homogeoneous polynomial in $\Bbb Z[W,X,Y,X]$ that contains only coefficients (which may be $0$) from $W^4,X^4,Y^4,Z^4,W^2X^2,W^2Y^2,X^2Z^2,Y^2Z^2,WXYZ$ that factorizes into unequal ...
0
votes
0answers
17 views

Composition of polynomial and multiplicative is multiplicative .

I made the following problem a while ago but I can't solve it (also I don't think it's extremely hard ) : Let $f$ be a non-constant completely multiplicative function over $\mathbb{Z}$ . Assume ...