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.
2,832 questions
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Does a quartic have four rational roots if its resolvent cubic has three rational roots?
Let
$$f(x)=ax^4+bx^3+cx^2+dx+e$$
using Farrari mathod,one associate to resolvent cubic
$$R(y)=py^3+qy^2+ry+s$$
It is well known that If the quartic $f(x)$ has four rational roots, then the ...
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Pólya-Szegő convexity theorem for level lines
I am looking for a precise reference for the following result, usually cited as the Pólya-Szegő Convexity Theorem: "If all the zeros of the polynomial $f(z)$ lie on the real axis, then every ...
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Zeros of polynomials associated to finite quivers
Let $Q$ be a finite connected quiver with path algebra $KQ$.
Let $C$ be the Cartan matrix of $KQ$ and $M:=C^{-1}+(C^{-1})^T$, the symmetric matrix correesponding to the symmetric Euler form of Q (up ...
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Is there an extension of Lindström's theorem to base $b>2$?
B.Lindström (1997) proved that for any polynomial $p(x) \in \mathbb{Z}[x]$ with positive leading coefficient and degree $h>1$,
$$\limsup\limits_{n\to\infty}\frac{s_2(p(n))}{\log_{2}n}=h$$
where $...
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Conjecture: linear recurrences with constant coefficients satisfy non-linear irreducible polynomial [closed]
Working over the integers, probably any field will do.
Let $a(n)$ be linear recurrence with constant coefficients:
$a(n)=c_1 a(n-1)+c_2 a(n-2)+\cdots +c_d a(n-d)$.
Conjecture 1 There exist integers $k,...
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Behavior of univariate polynomial rings over fields modulo powers of irreducible polynomials
In the recent preprint On some local rings, Mohamad Maassarani looks at the following question.
Given a field $\mathbb{k}$, two irreducible polynomials $P_1, P_2\in\mathbb{k}[x]$ and an integer $n>...
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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 ...
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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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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 ...
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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
$$ H_d(m) := \left( \frac{B_{i+j+1}(m)}{i+j+1} \right)...
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Lexicographically maximal vanishing sums of $n$-th roots of unity
Let $n$ be a positive integer with $6\mid n$, and let
$$
\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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