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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.
4
votes
Accepted
Solving a recursion for polynomials defined by a matrix product
Your polynomial is precisely
$$
\sum_{k_1+2k_2+\cdots+nk_n=n}\binom{k_1+\cdots+k_n}{k_1,\ldots,k_n}X_1^{k_1}\cdots X_n^{k_n}.
$$
The proof is straightforward by induction: you have
$$
p_n(X)=\sum_{i= …
- 16.9k
2
votes
Accepted
Ideal membership and change of fields
The ideal membership question is completely algorithmic, and indeed you can solve is using the reduced Gröbner basis of your ideal (for any ordering of your choice).
The reduced Gröbner basis of $(f_1 …
- 16.9k
5
votes
Is there a $3$-commutative algebra?
I have just thought of another interesting example, which is a bit peculiar, in that the 3-commutativity property arises for a natural subspace of an algebra which itself does not satisfy any identity …
- 16.9k
1
vote
Ideals invariant under translation of variables
You might be interested in "Gröbner bases of ideals invariant under endomorphisms", by Vesselin Drensky and Roberto La Scala, Journal of Symbolic Computation
(2006), Issue 7, Pages 835-846. In your sp …
- 16.9k
5
votes
Is there a $3$-commutative algebra?
This variety of algebras is well studied. In particular, the description of the underlying vector space of the free algebra in this variety follows immediately from
"A note on the T-ideal generated by …
- 16.9k
11
votes
Solve this sextic
The RHS only depends on $n(n+1)$, specifically it can be written as
$$
4a\left(8(n(n+1))^3+6(n(n+1))^2+n(n+1)+\frac12\right)+4bn(n+1)+2c,
$$
so you have to use the standard formulas to solve a cubic e …
- 16.9k
4
votes
Accepted
An $n$ eigenvalue multiplicity
Algebraic multiplicity $n$ means that we have the equality of polynomials
$$
\det(t I_n -a_1A_1+\cdots+a_nA_n)=(t-\lambda)^n
$$
for some $\lambda$. …
0
votes
Accepted
Transcendent basis for the field of multisymmetric functions
Deciding positivity of multisymmetric polynomials. Journal of Symbolic Computation 74 (2016), 603-616. …
- 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
2
votes
Jack polynomials and the Witt algebra
Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials. Comm. Math. Phys.
Volume 174, Number 2 (1995), 447-455.
B. Feigin, M. Jimbo, T. Miwa, E. Mukhin. … A differential ideal of symmetric polynomials spanned by Jack polynomials at rβ = -(r=1)/(k+1). International Mathematics Research Notices, Volume 2002, Issue 23, 2002, Pages 1223–1237. …
- 16.9k
3
votes
Accepted
A commutative variant of the exterior algebra
For $k=\infty$, this algebra appears when studying integrable representations of level 1 of the Lie algebra $\widehat{\mathfrak{sl}}_2$, see, for example, discussion in Section 2 of
A. V. Stoyanovs …
- 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
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
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
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 …
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