# 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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### 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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### 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 ...
357 views

### Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial?

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\setminus\{0,1\}.$ Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow from ...
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### 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 $\Frac(R)$ ...
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### Why do these two integrals use roots of reciprocal polynomials?

There is the nice integral by V. Reshetnikov, $$\int_0^1\frac{dx}{\sqrtx\ \sqrt{1-x}\ \sqrt{1-x\,\alpha^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{3}\;\alpha}\tag1$$ also discussed in this post. By ...
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### Number of distinct roots between complex roots among three polynomials

I want to prove that any three relatively prime polynomials $A, B, C \in \mathbb C[X]$ verifying $A+B+C=0$ have at least $1+\max(\deg A, \deg B, \deg C)$ distinct roots in total among each other. I ...
### $a$ and $b$ are solutions of $\frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0$, $a+b=?$
$a$ and $b$ are solutions of $$\frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0$$ What is $a+b=?$  Are there better approaches than the one below? Solution: ...
### $P(x,y) = n$ iff $n$ is NOT a perfect square
Does there exist a two variable polynomial $P(X,Y)$ with integer coefficients such that a positive integer $n$ is a perfect square iff there do not exists a tuple $(X,Y)$ of positive integers such ...