# Tagged Questions

Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-...

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### Making a multivariate polynomial monic in one of its variables

I apologise in advance for the general nature of this question. Suppose we have a non-commutative ring $R$ that is relatively well-behaved as non-commutative rings go (I was thinking of $R$ being the ...
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### Polynomial ring operations on $\mathbb{Z}$

I have asked this on Math Stack Exchange but without answers: The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables): ...
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### Polynomial with subset of critical points and values prescribed

Motivated by this question I am motivated to pose the following question: Given $k$ points $(x_1, y_1), \ldots (x_k, y_k)$ with (WLOG) $x_i < x_{i+1}$, can we find a polynomial $p(x)\in\Bbb R[x]$ ...
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### Metrics on the group of unimodular polynomial matrices

The group of unimodular matrices $\mathbb{U}[s]^{n\times n}$ is given by the set of $n\times n$ square (real) matrix-valued polynomials $\mathbb{R}[s]^{n\times n}$ which admit a polynomial inverse. ...
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### Coefficients of factors of $x^n-1\in\mathbb{Q}[x]$

If you factor $x^n-1\in\mathbb{Q}[x]$, then for $n\leq 104$ the coefficients of the factors are in $\{-1, 0, 1\}$. (This is not true for $n=105$, however). Let $U$ be the set of positive integers $n$ ...
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### Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$ for all positive integers $n$

Let $n$ be a positive integer greater than $1$, and define the polynomial $$p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$$ Is $p_{n}(x)$ irreducible in $\mathbf{Q}[x]$? I can show it when $n$ is a ...
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### On property of monic polynomial with integer coefficients

For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have $$\textrm{inf}(f(x)) > 0 \implies \textrm{inf}(f(x)) \geq \frac{3}{4} .$$ Could we generalize this (for ...
$\newcommand\KK{\mathbb{K}}$Let $\KK$ be any field and $f\in\KK[x_1,\dotsc,x_n]$ be a polynomial. Its support $S_f$ is the set $\{(e_1,\dotsc, e_n) : x_1^{e_1}\dotsb x_n^{e_n}$ has a nonzero ...