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Results tagged with polynomials
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user 38620
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.
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Why can't this polynomial vanish except when $x+y=0,xy= 0$? [closed]
Show that for any $x, y \in \mathbb R$ with $x + y \neq 0,xy\neq 0$
$$p(x,y) := x^6-2 x^5 y+2 x^5-x^4 y^2-2 x^4 y+x^4+4 x^3 y^3+2 x^3 y-x^2 y^4-4 x^2 y^3-4 x^2 y^2+2 x^2 y-2 x y^5+6 x y^4+2 x y^3+y^6 …
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0
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272
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For which constant $d$ this polynomial is reducible over $\mathbb Q$? [closed]
Let $$f(x)=(x-1)^2(x-2)^2(x-3)^2\cdots(x-2013)^2+2014\tag{1}$$
Prove or disprove: $f(x)$ is reducible over $\mathbb Q$.
See :http://www-irma.u-strasbg.fr/~bugeaud/travaux/PolyaType.pdf
Ques …
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6
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1
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Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$ for all positive integers $n$
Let $n$ be a positive integer greater than $1$, and define the polynomial $$p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$$
Is $p_{n}(x)$ irreducible in $\mathbf{Q}[x]$?
I can show it when $n$ is a pr …
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4
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1
answer
316
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maybe Faulhaber polynomial $S_{k}(x)=0$ have only rational roots $0,-\frac{1}{2},-1$
Faulhaber polynomial of order $p \in \Bbb{N}$ is defined as the unique polynomial of degree $p+1$ satisfying
$$ S_{p}(n) = \sum_{k=1}^{n} k^p $$
for $n = 1, 2, 3, \cdots$. For example,
\begin{align*}
…
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8
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0
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How prove this polynomial inequality from a book
Question:
Let $$f(X)=(X-x_{1})(X-x_{2})\cdots (X-x_{n})$$ be an irreducible polynomial over the field of rational numbers,with integer coefficients and real zeros.
Prove that
$$\prod_{1\le i<j\le n}| …
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2
votes
1
answer
329
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Determine whether a system of polynomials with real coefficients has a real solution?
Today my students asked me the following problem：
Define polynomials $P_j$ with real coefficients
$$P_{j}=\sum_{i_{1},\cdots,i_{s}}a^{(j)}_{i_{1}\cdots i_{s}}X^{i_{1}}_{1}\cdots X^{i_{s}}_{s}, \qquad … But for a system of real polynomials to have a real solution, I can't find any similar results in the literature. Maybe this result is old? Thank you for you help. …
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7
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1
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Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$
I conjecture that the answer is $|[x^p]T_{n}(x)|$, where the $T_{n}(x)$ are the Chebyshev polynomials. Can we find the closed form for it? Thanks …
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38
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4
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A family of polynomials whose zeros all lie on the unit circle
I had posted the following problem on stack exchange before.
Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the polynomi …
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