Questions tagged [polynomials]
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-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
2,826 questions
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Sequences of irreducible polynomials
During some digging of mine, I once found the following recursively defined family of polynomials: $P_0=P^2+2; P_{k+1}=P_k^2-2$. Using them one can show with purely algebraic means that the 2-adic ...
16
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2
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Irreducibility of a family of integer polynomials
"Let $f(x)$ be a polynomial of degree at least 2 with $f(\mathbb{N})\subset \mathbb{N}$. Then the set of natural numbers $n$ such that $f(x)-n$ is reducible over $\mathbb{Q}$ has density 0."
...
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Global to local solutions of $x^n-y=0$
Let $n,y$ be positive integers that are greater than 1. Suppose that $x^n=y$ does not have a solution in positive integers. Then is it true that for infinitely many primes $p$ there are no solutions $...
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1
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Polynomial taking only 0 and 1 values at many consecutive integers
The interpolation already gives a polynomial $f(x)$ of degree at most $n$ that makes $(f(0),f(1),\dotsc, f(n))$ attain any real sequence in $\mathbb{R}^{n+1}$.
I'm curious about the following:
Is ...
- 353
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Solving a large system of polynomial equations over $\mathbb{F}_2$
In my work I commonly encounter large systems of polynomial equations for which it would be useful to know if there is a nontrivial solution over $\mathbb{F}_2$ (and if so, to find such a solution).
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Minimal number of variables needed to parametrize $\operatorname{SL}_2(\mathbb{Z})$
Recently I am studying the paper
Leonid Vaserstein, Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Annals of Mathematics 171 issue 2 (2010) pp. 979–1009, ...
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Is this Hankel matrix involving Bernoulli polynomials positive definite?
Let $(B_n(X))_{n \ge 0}$ denote the sequence of Bernoulli polynomials. For any couple of integers $(d,m) \in \mathbb{N}$, define the Hankel matrix
$$ H_d(m) := \left( \frac{B_{i+j+1}(m)}{i+j+1} \right)...
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0
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Lexicographically maximal vanishing sums of $n$-th roots of unity
Let $n$ be a positive integer with $6\mid n$, and let
$$
\zeta := e^{2\pi i / n}.
$$
For a given integer $m\in\{2,\dots,n-2\}$, consider subsets
$$
A \subset \{0,1,\dots,n-1\}
$$
of size $|A|=m$ ...
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Divisibility relations among degrees of irreducible factors of a binomial
Suppose $K$ is an algebraic number field, and $a \in K$. Let $n$ be a positive integer. The polynomial $t^n - a \in K[t]$ splits as a product of irreducible factors of degrees $d_1, \dots, d_r$.
Is it ...
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Characterize nonzero integers via a polynomial in two variables
In a paper published in 1985, Shih-Ping Tung observed that an integer $m$ is nonzero if and only if $m=(2x+1)(3y+1)$ for some $x,y\in\mathbb Z$. In fact, we can write a nonzero integer $m$ as
$\pm3^a(...
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Linear-Disjointness of the field obtained upon iterated pre-images
Suppose $K$ is a number field and we have finitely many elements $\alpha_{1}, \ldots, \alpha_{k} \in K$ and a polynomial $f(x)\in K[x]$ of degree $\geq 2$. Given $m \geq 1$, we define $K_{m}(\alpha_{i}...
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Zeros of the partial sums $\sum_{k=0}^n (-1)^k/(z-k)$
let
$$
D_n(z)=\sum_{k=0}^n \frac{(-1)^k}{z-k}
=\frac{P_n(z)}{Q_n(z)}, \qquad
Q_n(z) = \prod_{k=0}^n (z-k).
$$
We have $\deg P_n = n$ ($n$ even), and $\deg P_n = n-1$ ($n$ odd). Moreover, using the ...
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How close can general algebraic curves get to rational curves?
Let $K$ be a field (one may assume that $K$ has characteristic zero and is algebraically closed, if this helps), and let $f \in K[x,y]$. Suppose that the curve $C_f : f(x,y) = 0$ is not a rational ...
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Many degree $d$ nilpotent elements of quotients of polynomial rings and non-vanishing product
Generalization of this question.
Let $n$ be positive integer and $f_1(x_1,...,x_k), \dots, f_m(x_1,...x_k)$
polynomials with integer coefficients.
Let $K=\mathbb{Z}/n\mathbb{Z}[x_1,...,x_k]/\langle ...
- 25.7k
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1
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A question about positive polynomials
Are there some $P,Q \in \mathbb R_+[x]$ with $(x+10)^{2025}=(x+2025)^2P(x)+(x+2024)^2Q(x)$ ?
PS : the AI give an negative answer in the case $(x+1)^{2025}$
I have posted the question here (*), but no ...
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