The minimal number of invariants needed to generate ${\mathbb Z}[x_1,...,x_n]^{{\mathbb Z}}_n}$ Z}_n}$ has been considered by a number of authors including Erdos, Dixmier and Kac. (see the references in [John C. Harris and David L. Wehlau Non-Negative Integer Linear Congruences, Indagationes Mathematicae 17 No. 1 (2006) 37-44]. It is easily seen to be bounded below by the number of partitions of n, ${\mathcal P}(n)$. Dixmier produced a number of papers giving the asymptotic behavior of this number as a function of $n$. The results in the above Harris-Wehlau paper are completed by using the main result in [Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551]. These two papers combine to show that the number of homogeneous generators of this ring of invariants of degree $k$ is exactly $\phi(n){\mathcal P}(n-k)$ if $k \geq \lfloor n/2\rfloor + 2$ (here $\phi$ is Euler's totient function). Surprisingly (at least to me) much less is known about the number of generators in lower degrees.
|
4 | latex fix | ||
|
|
||||
|
|
3 | corrected spelling error | ||
|
The minimal number of invariants needed to generate ${\mathbb Z}[x_1,...,x_n]^{{\mathbb Z}}_n}$ has been considered by a number of authors including Erdos, Dixmier and Kac. (see the references in [John C. Harris and David L. Wehlau Non-Negative Integer Linear Congruences, Indagationes Mathematicae 17 No. 1 (2006) 37-44]. It is easily seen to be bounded below by the number of partitions of n, ${\mathcal P}(n)$. Dixmier produced a number of papers giving the asymptotic behavior of this number as a function of $n$. The results in the above Harris-Wehlau paper are completed by using the main result in [Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551]. These two papers combine to show that the number of homogeneous generators of this ring of invariants of degree $k$ is exactly $\phi(n){\mathcal P}(n-k)$ if $k \geq \lfloor n/2\rfloor + 2$ (here $\phi$ is Eulor's Euler's totient function). Surprisingly (at least to me) much less is known about the number of generators in lower degrees. |
||||
|
|
2 | added 12 characters in body | ||
|
The minimal number of invariants needed to generate ${\mathbb Z}[x_1,...,x_n]^{{\mathbb Z}}_n}$ has been considered by a number of authors including Erdos, Dixmier and Kac. (see the references in [John C. Harris and David L. Wehlau Non-Negative Integer Linear Congruences, Indagationes Mathematicae 17 No. 1 (2006) 37-44]. It is easily seen to be bounded below by the number of partitions of n, ${\mathcal P}(n)$. Dixmier produced a number of papers giving the asymptotic behavior of this number as a function of $n$. The results in the above Harris-Wehlau paper are completed by using the main result in [Pingzhi Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551]. These two papers combine to show that the number of homogeneous generators of this ring of invariants of degree $k$ is exactly $\phi(n){\mathcal P}(n-k)$ if $k \geq \lfloor n/2\rfloor + 2$ (here $\phi$ is Eulor's totient function). Surprisingly (at least to me) much less is known about the number of generators in lower degrees. |
||||
|
1 |
|
||
