# Questions tagged [polynomials]

For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.

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### If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
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### Does there exist a polynomial $P(x,y)$ which detects all non-squares?

Problem. Does there exist a two-variable polynomial $P(x, y)$ with integer coefficients such that a positive integer $n$ is not a perfect square if and only if there is a pair $(x, y)$ of positive ...
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### How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and four ...
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### When is a polynomial contained in the ideal generated by its partial derivatives?

Let $R = k[x_1,\dots,x_n]$ be a multivariate polynomial ring over a field $k$ of characteristic zero, and let $f\in R$. Is there an easy-to-test necessary and sufficient condition on $f$ such that $f$...
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### Is $z=e^{\frac{1}{\log(x)}}$ a solution to an algebraic equation?

Is $z=e^{\frac{1}{\log(x)}}$ with $x\in\Bbb Q~\cap(0,1)$ a solution to an algebraic equation? Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow ...
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### A Polynomial Formed from the Roots of Another Polynomial ad infinitum

Let $P(x)$ be a monic polynomial of degree $d$ with complex coefficients. Let $r_1(P),r_2(P),\dots, r_d(P)$ denote the set of roots, ordered so that $|r_1(P)| \leq |r_2(P)|\leq\dots\leq |r_d(P)|$. ...
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### Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
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### If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$

I'm trying to understand a proof of the following theorem (from section II of Hall's paper An Isomorphism Between Linear Recurring Sequences and Algebraic Rings): If $F(a_1, \ldots, a_k)$ is a ...
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### On the continuation of a polynomial

This exrcise is from the first section of Marden: Exercise 12. Let the interior of a piecewise regular curve $C$ contain the origin $\cal O$ and be star-shaped with respect to $\cal O$. If the ...
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### Prove that $\Phi_{420}(69) > \Phi_{69}(420)$

Let $\Phi_n(x)$ denote the nth cyclotomic polynomial. Prove that $\Phi_{420}(69) > \Phi_{69}(420)$. Observe that if $\phi$ denotes the Euler-phi function, \begin{align} \phi(420) &= (2^2-2) \...
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### Quantify the similarity between a polynomial roots and the roots of its derivatives

On $\mathbb{C}[X]$, many theorems and conjectures deal with relations between a polynomial roots and the roots of its derivatives. When looking at a graph, the derivative roots distribution somewhat ...
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### Are circle-tangent polynomials always either odd or even?

By saying circle-tangent polynomial I mean a polynomial of degree $N>2$ that lies mostly inside a unit circle except for the two tails exiting the circle and going to infinity and is tangent to the ...
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### To get all open sets in $\mathbb{R}^n$ with a unique polynomial

At the beginning of the book Tame topology and o-minimal structures, by Lou van den Dries, it said that all the open sets of $\mathbb{R}^n$ can be obtained from a single polynomial equation. More ...
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### Is theory of equations a dead field today?

Is theory of equations a dead field today? By theory of equations I mean, specially, the study of polynomials and solving algebraic equations through radicals. There seem to be very few journals on ...
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### Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$. Definition: A polynomial $f$ is $d$-SOS if ...
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### Sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$

What are some sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$ with $\alpha , \beta$ algebraic over $\mathbb{Q}$? We know that, for ...
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If $f(z) \in \mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,... • 161 8 votes 0 answers 147 views ### If prime$p=a_n10^n+a_{n-1}10^{n-1}+\ldots+a_110+a_0$then$f(x)=a_nx^n+\ldots+a_0$is irreducible in$\mathbb{Z}[x]$I have been trying to solve this problem on my own for four days now, and I cannot figure out how to prove it: If we express a prime$p$in base$10$as$$p= a_m10^m+a_{m-1}10^{m-1}+\ldots +a_110+a_0,... • 267 8 votes 0 answers 252 views ### Domains such that$R[X] \cong S[X]$but$\mathrm{Frac}(R)[X] \not \cong \mathrm{Frac}(S)[X]$Is it possible to find two integral domains$R,S$such that$R[X] \cong S[X]$but$\mathrm{Frac}(R)[X] \not \cong \mathrm{Frac}(S)[X]$?$\renewcommand{\Frac}{\mathrm{Frac}}$Here$\Frac(R)$... • 23.8k 8 votes 0 answers 231 views ### Fixed point of a polynomial mapping - what's the relation between the two views Let$\sigma : \Bbb{C}^3 \to \Bbb{C}^3$be a polynomial mapping. Let$P:= \Bbb{C}[x,y,z]$denote the space of polynomial in 3 variables. Then$\sigma$induces a (linear) mapping$\tilde{\sigma} : P\...
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Given two integers ($n$, $m$), what is the smallest number of terms that could result from the product of two polynomials with $n$ and $m$ non-zero terms respectively? That is, what is the smallest ...
We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...