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.

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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 ...
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Connected components of pseudospectra

In this Article, page 5 Theorem 2.3 ,what is connected components of pseudospectra of matrix polynomial?
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What is connected components of pseudospectra of matrix polynomial? . [on hold]

What is connected components of pseudospectra of matrix polynomial? Please see this link
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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 ...
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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 ...
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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 ...
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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. ...
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Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\alpha)\cdot g(x)$, where $deg(p)=n-1$, $\alpha \leftarrow \mathbb{Z}_p$. We evaluate $P$ at some $x_i$ values. So we get $(x_1, y_1),...,(x_n, y_n)$, where $P(x_i)=y_i$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $n-2$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) = ...
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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 ...
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How to obtain a specified set of coefficients of a multivariate polynomial

First of all, I am a physicist without any training in abstract algebra or related fields which means I will probably not be able to ask this question using specific definitions from these fields. ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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, ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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Why does $\frac{x^n}{n^x}$ stop growing at the approximate value of $\pi (n)$?

I noticed while playing around with these functions that $n^x$ will start slow and then speed up really fast in its growth rate. While $x^n$ grows more slowly, but faster than $n^x$ at the start. ...
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Finding coefficients of $x^n$ and $x^{n+r}$ in an expansion

I have to find the coefficients of $x^n$ and $x^{n+r}$ $(1 < r < n)$ in the expansion of: $$(1 + x)^{2n} + x(1 + x)^{2n - 1} + x^2(1 + x)^{2n - 2} + ... + x^n(1 + x)^n$$ How do I solve it?
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Legendre symbol identity

I try to solve the following problems ($p$ is an odd prime) Find the sum $$\sum_{a=1}^{p-1}a \cdot \left (\frac{a}{p} \right)$$ Find the sum $$\sum_{a=1}^{p-1} 2^a \cdot \left (\frac{a}{p} \right)$$ ...
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Approach for optimization problem with polynomial constraints?

I have a problem where the objective function is linear and constraints have polynomials (in one variable). So, my question is what are the main approaches to this issue? I can construct a small ...
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How to factor $ s^2LC + sRC + 2$

or $$ s^2+s\frac{R}{L}+\frac{2}{LC}=0 $$ Is there any way? I can't find out. Thanks in advance.
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How do I solve an equation like this?

How do I solve following equation for $X$: $$ AX^n + BX^{n-1} + CX^{n-2} + \dotsb + YX + Z = 0, $$ where $A,B,C,\dotsc,Z,n$ are known?
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Sum of digits modulo a polynomial

I made the following problems a while ago but I can't solve them (though I don't think it's too hard) 1.Let $s(n)$ be the digits sum of $n$. Let also $f(n)$, $g(n)$ $\in Z[X]$ . Assume that: ...
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Durand-Kerner with derivative in denominator

The correction term for Durand-Kerner root finding method is $w_k = -\frac{f(z_k)}{\prod_{j\not=k}(z_k - z_j)}$ Wikipedia Talk page mentions that it is also possible to use derivative in ...
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Contest problem in functional equations.

Let n be a positive integer with $f(n)= 1! +2! +3!+... +n!$ and P(x), Q(x) be polynomials in $x$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$, then which of the options is/are ...
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Conditions for a unique root of a fifth degree polynomial

Fifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient ...
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Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ...
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Questions from an olympiad on number theory [closed]

The sum of the infinite series: $$ \frac{1}{2} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +....$$ I am able to find the general term ...