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Results tagged with polynomials
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user 70464
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.
4
votes
1
answer
200
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$2$-adic order bound for $P(x)$
Let $P(x) \in \mathbb{Z}[x]$ be a monic polynomial of degree $n$, with no integer roots, such that $\upsilon_2(P(m))$ can assume $n$ different positive values, where $m \in \mathbb{Z}$ and $\upsilon_ …
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1
vote
1
answer
369
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Construction of an irreducible polynomial on $\mathbb{Z}[x]$
Given a positive integer $m>1$, prove that there exists a polynomial of integer coefficients $P(x)$ that is irreducible on $\mathbb{Z}[x]$, has degree $2018$ and has a real root $r$ satisfying $m \mi …
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30
votes
3
answers
2k
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All polynomials are the sum of three others, each of which has only real roots
Volume 64, Number 2, 1958, as a Research Problem, if a Hurwitz polynomial with real coefficients (i.e. all of its zeros have negative real parts) can be divided into the arithmetic sum of two or three polynomials … I would like to know if the following problem is known and how it can be solved:
Can any polynomial with complex coefficients and degree $n$ be divided into the arithmetic sum of three complex polynomials …
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6
votes
1
answer
311
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$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer
Does anyone know if the following problem has ever been studied?
Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$
where $n$ is a posit …
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10
votes
1
answer
343
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Finding $q(x)$ such that $p(q(x))$ is reducible over $\mathbb{Q}[x]$
Let $p(x) \in \mathbb{Z}[x]$, such that $\deg (p) \ge 3$.
Can we always find $q(x) \in \mathbb{Z}[x]$, such that $\deg (q) < \deg(p)$ and $p(q(x))$ is reducible over $\mathbb{Q}[x]$?
Is there a …
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20
votes
1
answer
751
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Minimum value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$
.+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.
This question was proposed (problem A.611)
some time ago at KoMaL.
The minimal values for $n=0,1,2,3$ are $3,5,14,324/5$. …
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29
votes
6
answers
2k
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$m$-fold composite $p^{(m)}(x) \in \mathbb{Z}[x]$ implies $p(x) \in \mathbb{Z}[x]$
Let $p(x)$ be a polynomial, $p(x) \in \mathbb{Q}[x]$, and $p^{(m+1)}(x)=p(p^{(m)}(x))$ for any positive integer $m$.
If $p^{(2)}(x) \in \mathbb{Z}[x]$ it's not possible to say that $p(x) \in \mathbb{Z …
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-1
votes
1
answer
141
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If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $...
Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer.
In a previous post I asked if $p_n(a,b)$ was a rationa …
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