Title: SETS WITH EVEN PARTITION FUNCTIONS AND CYCLOTOMIC NUMBERS
Abstract: Let $P\in \mathbb{F}_{2}[z]$ be such that $P(0)=1$ and degree $(P)\geq 1$ . Nicolas et al. [‘On the parity of additive representation functions’, J. Number Theory 73 (1998), 292–317] proved that there exists a unique subset ${\mathcal{A}}={\mathcal{A}}(P)$ of $\mathbb{N}$ such that $\sum _{n\geq 0}p({\mathcal{A}},n)z^{n}\equiv P(z)~\text{mod}\,2$ , where $p({\mathcal{A}},n)$ is the number of partitions of $n$ with parts in ${\mathcal{A}}$ . Let $m$ be an odd positive integer and let ${\it\chi}({\mathcal{A}},.)$ be the characteristic function of the set ${\mathcal{A}}$ . Finding the elements of the set ${\mathcal{A}}$ of the form $2^{k}m$ , $k\geq 0$ , is closely related to the $2$ -adic integer $S({\mathcal{A}},m)={\it\chi}({\mathcal{A}},m)+2{\it\chi}({\mathcal{A}},2m)+4{\it\chi}({\mathcal{A}},4m)+\cdots =\sum _{k=0}^{\infty }2^{k}{\it\chi}({\mathcal{A}},2^{k}m)$ , which has been shown to be an algebraic number. Let $G_{m}$ be the minimal polynomial of $S({\mathcal{A}},m)$ . In precedent works there were treated the case $P$ irreducible of odd prime order $p$ . In this setting, taking $p=1+ef$ , where $f$ is the order of $2$ modulo $p$ , explicit determinations of the coefficients of $G_{m}$ have been made for $e=2$ and 3. In this paper, we treat the case $e=4$ and use the cyclotomic numbers to make explicit $G_{m}$ .
Publication Year: 2016
Publication Date: 2016-03-14
Language: en
Type: article
Indexed In: ['crossref']
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