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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.
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
4
votes
Accepted
Is a polynomial with 1 very large coefficient irreducible?
OK, about your second question. Let's consider the polynomial $x^n+2(x^{n-1}+\ldots+x^2+x)+4$. I claim that this polynomial is almost good for your purposes: if we permute all coefficients except for …
- 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
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:
$$
x_i=\frac{(k-2i)\sqrt{k}+(2 …
- 16.9k
10
votes
Looking for ways how to calculate $\Phi_n(i)$
I suggest to use the following properties of cyclotomic polynomials:
for $p$ prime, $k>1$, we have $\Phi_{mp^k}(x)=\Phi_{mp}(x^{p^{k-1}})$;
for $p$ prime and $m$ coprime with $p$, we have $\Phi_{mp …
- 16.9k
3
votes
Accepted
Bound the degree of the generator of polynomial ring
Without homogeneity assumptions, the intersection need not be finitely generated. This is discussed in http://arxiv.org/abs/1301.2730. (See also http://www.emis.de/journals/BAG/vol.43/no.2/b43h2bay.pd …
- 16.9k
2
votes
A combinatorial formula involving the necklace polynomial
The LHS of your formula can be rewritten as
$$
\sum_{k\ge 1, d\mid k}\left[\frac{n}{k}\right]\mu(k/d)X^d,
$$
which after rearranging terms becomes
$$
\sum_{d\ge 1}X^d\sum_{s\ge 1}\mu(s)\left[\frac{ …
- 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
1
vote
Relation between degree of root of determinant polynomial and rank of the matrix
From your more general question I infer that you want to look at the coset of your matrix in the quotient (not at evaluation at specific $x_1,\ldots,x_n\in\mathbb{F}_q$).
Without loss of generality, …
- 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
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
1
vote
Ideals invariant under translation of variables
You might be interested in "Gröbner bases of ideals invariant under endomorphisms", by Vesselin Drensky and Roberto La Scala, Journal of Symbolic Computation
(2006), Issue 7, Pages 835-846. In your sp …
- 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
5
votes
Signed factors of harmonic polynomials
.
$$
For $n=3$ it looks plausible too, since Legendre polynomials appear in explicit formulas, and maybeone can play with the known fact on their roots. … Positive definiteness of polynomials is a subtle thing (cf. Hilbert's 17th problem etc.)... …
- 16.9k
4
votes
Accepted
An $n$ eigenvalue multiplicity
Algebraic multiplicity $n$ means that we have the equality of polynomials
$$
\det(t I_n -a_1A_1+\cdots+a_nA_n)=(t-\lambda)^n
$$
for some $\lambda$. …