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Results tagged with polynomials
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user 1306
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.
56
votes
Accepted
Minimal polynomial of cos(π/n)
The minimal polynomial of $\cos(2\pi/n)$ (by William Watkins and Joel Zeitlin, The American Mathematical Monthly
Vol. 100, No. 5 (May, 1993), pp. 471-474) has full clarity on this matter (just take th …
- 16.9k
25
votes
Relations between sums of powers
Surely there are many: these are all polynomials in one variable, so every two of them are algebraically dependent because of the transcendence degree argument :-)
However, I am sure that this is not … such a combination is clearly unique - you yourself observed that they form a basis), and because of the type of symmetry it possesses, it is actually a combination of $P_1$ and $P_3$ (because other polynomials …
- 16.9k
16
votes
Accepted
Cyclotomic polynomials.
If this were true, then you would prove that $\Phi_m(x)$ and $\Phi_n(x)$ are coprime after reduction modulo $p$, which is far from true. For instance, $\Phi_4(x)=x^2+1$ and $\Phi_2(x)=x+1$ are not cop …
- 16.9k
14
votes
The number of irreducible polynomials over ${\mathbb F}_p$
irreducibles uniquely, hence we have a formal power series equality
$$
\frac{1}{1-qt}=\prod_{k\ge 1}\frac{1}{(1-t^k)^{M_k(q)}}
$$
(indeed, the coefficient of $t^n$ on the left is the number of monic polynomials …
- 16.9k
13
votes
Accepted
What are retracts of polynomial rings?
Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article
Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inve …
- 16.9k
13
votes
1
answer
1k
views
Irreducibility of Schur polynomials
A natural question covering both this and this question would be
Let $n>2$. Describe Young diagrams $\lambda$ with at most $n$ nonempty rows (or equivalently non-increasing sequences $\lambda=(\lamb …
- 16.9k
11
votes
Solve this sextic
The RHS only depends on $n(n+1)$, specifically it can be written as
$$
4a\left(8(n(n+1))^3+6(n(n+1))^2+n(n+1)+\frac12\right)+4bn(n+1)+2c,
$$
so you have to use the standard formulas to solve a cubic e …
- 16.9k
10
votes
Looking for ways how to calculate $\Phi_n(i)$
I suggest to use the following properties of cyclotomic polynomials:
for $p$ prime, $k>1$, we have $\Phi_{mp^k}(x)=\Phi_{mp}(x^{p^{k-1}})$;
for $p$ prime and $m$ coprime with $p$, we have $\Phi_{mp …
- 16.9k
10
votes
Is a polynomial with 1 very large coefficient irreducible?
Two other remarks supporting this comment: polynomials $x^n-N^n$ suggest that you should not expect anything for the constant term, and polynomials $(x^2-Nx+1)(x^2+Nx+1)=x^4+(2-N^2)x^2+1$ show that $k= …
- 16.9k
7
votes
Variants of Eisenstein irreducibility
A good source to learn about it is Prasolov's book on polynomials: http://tinyurl.com/prasolov - see page 53, Dumas' theorem (and a bit before this theorem). …
- 16.9k
6
votes
Accepted
Existence of solutions of a polynomial system
The solutions described via the link http://winvector.github.io/freq/explicitSolution.html (posted in one of the earlier answers) can be given by the following formula:
$$
x_i=\frac{(k-2i)\sqrt{k}+(2 …
- 16.9k
5
votes
Signed factors of harmonic polynomials
.
$$
For $n=3$ it looks plausible too, since Legendre polynomials appear in explicit formulas, and maybeone can play with the known fact on their roots. … Positive definiteness of polynomials is a subtle thing (cf. Hilbert's 17th problem etc.)... …
- 16.9k
5
votes
Is there a $3$-commutative algebra?
This variety of algebras is well studied. In particular, the description of the underlying vector space of the free algebra in this variety follows immediately from
"A note on the T-ideal generated by …
- 16.9k
5
votes
Is there a $3$-commutative algebra?
I have just thought of another interesting example, which is a bit peculiar, in that the 3-commutativity property arises for a natural subspace of an algebra which itself does not satisfy any identity …
- 16.9k
4
votes
Degree of sum of algebraic numbers
Irreducibles and the composed product for polynomials over a finite field. MR0893074 (89g:11118)). …
- 16.9k