受限分拆函数最小周期猜想的反例
Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions
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中文总结 AI 辅助
针对受限分拆函数系数最小周期的Beck-Sam-Woods猜想，推导系数的精确单位根公式，证明周期的整除上下界均为紧的，并构造出该猜想的一族反例。
中文摘要 AI 辅助
对于正整数有限序列$\boldsymbol{a}=(a_1,\dots,a_n)$，受限分拆函数$q_{\boldsymbol{a}}(k)$表示方程$a_1x_1+a_2x_2+\cdots +a_nx_n=k$的非负整数解的个数，已被证明是次数为$n-1$的拟多项式。将$q_{\boldsymbol{a}}(k)$写为$\textstyle\big\sum_{j=0}^{n-1}c_j(k)k^j$，其中$c_j$为周期系数函数，记$b_m=\number\backslash\\#\backslash\{i:m\nmid a_i\backslash\}$。2008年，Beck、Sam和Woods猜想$c_j(k)$的最小周期为$\text{lcm}\backslash\{m:b_m>j\backslash\}$。本文推导了每个系数函数$c_j(k)$的精确单位根公式，该公式证明了猜想的整除性上界，同时也揭示了$c_j(k)$周期的一个下界，两个整除界都是紧的，由此我们构造了该猜想的一族反例。
英文摘要
For a finite sequence of positive integers $\boldsymbol{a}=(a_1,\dots,a_n)$, the restricted partition function $q_{\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\boldsymbol{a}}(k)=\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\#\{i:m\mid a_i\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\mathrm{lcm}\{m:b_m>j\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.
