We present two new algorithms for the computation of the q-integer linear decomposition of a multivariate polynomial. Such a decomposition is essential in the q-analogous world of symbolic summation, for example, describing the q-counterpart of Ore-Sato theory or determining the applicability of the q-analogue of Zeilberger's algorithm to a q-hypergeometric term. Both of our algorithms require only basic integer and polynomial arithmetic and work for any unique factorization domain containing the ring of integers. Complete complexity analyses are conducted for both our algorithms and two previous algorithms in the case of multivariate integer polynomials, showing that our algorithms have better theoretical performances. A Maple implementation is also included which suggests that our algorithms are also much faster in practice than previous algorithms.

中文翻译：

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多元多项式的高效 q 整数线性分解

我们提出了两种用于计算多元多项式的 q 整数线性分解的新算法。这种分解在符号求和的 q 类比世界中是必不可少的，例如，描述 Ore-Sato 理论的 q 对应部分或确定 Zeilberger 算法的 q 类比对 q 超几何项的适用性。我们的两种算法都只需要基本的整数和多项式算术，并且适用于任何包含整数环的唯一分解域。在多元整数多项式的情况下，我们的算法和之前的两种算法都进行了完整的复杂性分析，表明我们的算法具有更好的理论性能。