All Questions
14 questions
2
votes
0
answers
114
views
How many minimal relations are needed to obtain a Frobenius algebra?
Let $A_n:=K \langle x_1,x_2,...,x_n \rangle$ be the non-commutative polynomial ring in $n$-variables over the field $K$ and let $J=\langle x_1,...,x_n \rangle$ be the ideal spanned by the $x_i$.
An ...
- 26.5k
4
votes
1
answer
370
views
Determining when quotient of a polynomial ring is a Gorenstein ring
I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...
- 214
1
vote
0
answers
329
views
Outlier absences of monomials in a group of inversion partition polynomials
Revamped and updated on Sep 12, 2022:
Given the complex coefficients $a_n$ of some generic formal power, Taylor, Laurent or other series, say the ordinary generating functions (o.g.f.) $f(z) = z +a_1 ...
- 10.5k
9
votes
2
answers
790
views
Algebraic power series of finite order
Apologies if the question is too elementary/something well-known.
I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
- 24.2k
3
votes
0
answers
107
views
Do Frobenius algebras have a lattice basis and what lattices do appear?
Let $K$ be for simplicity be the field with two or three elements (or alternatively we could restrict to ideals containing only the field elements $-1$ or $1$ as coefficients).
A (commutative) ...
- 26.5k
3
votes
0
answers
86
views
$\mathbb Z$-torsion for some quadratically presented Lie rings
$\newcommand{\Z}{\mathbb{Z}}$
I asked this question on MSE but no answer so far, so I'm also asking it here.
Let $L$ be a Lie ring (a Lie algebra over $\Z$) with generators $x_1,\dots,x_n$ and ...
- 8,524
5
votes
1
answer
177
views
an algebra generated by some known series
Denote the e.g.f. for the number of (unordered) rooted labeled trees on $n$ nodes by
$$\Phi(x)=\sum_{n\geq1}\frac{n^{n-1}}{n!}x^n.$$
And, the related series $\Psi(x)=\sum_{n\geq1}\frac{n^n}{n!}x^n$. ...
- 43.2k
5
votes
0
answers
250
views
A question on symmetric functions
Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
- 7,249
18
votes
2
answers
2k
views
Can Schwartz-Zippel be formulated for commutative rings instead of fields?
The polynomials which occur in the Schwartz-Zippel lemma could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for $(1+x^n)=1+x^n(\operatorname{...
- 2,464
5
votes
3
answers
1k
views
Stanley-Reisner ring of a simplicial complex is a functor?
Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let $\...
- 1,589
18
votes
5
answers
2k
views
Is a complete homogeneous symmetric polynomial irreducible?
Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$.
$$ h_a=\text{ sum of all monomials of degree }...
- 446
1
vote
0
answers
138
views
Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
5
votes
0
answers
517
views
Monomial-type ideals in polynomial rings
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
- 805
3
votes
7
answers
4k
views
How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
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