We study three systems of equations, together with a way to count the number of solutions. One of the results was used in the recent computation of D(9), the others have potential to speed up existing techniques in the future.

翻译：我们研究了三个方程组及其解的计数方法。其中一项成果已应用于近期D(9)的计算实践，其余成果则具备提升现有技术效率的潜在价值。