This paper proposes an algorithm to reason on constraints expressed in terms of polynomials with integer coefficients whose variables take values from finite subsets of the integers. The proposed algorithm assumes that an initial approximation of the domains of variables is available in terms of a bounding box, and it recursively subdivides the box into disjoint boxes until a termination condition is met. The algorithm includes three termination conditions that allow using it for three related reasoning tasks: constraint satisfaction, enumeration of solutions, and hyper-arc consistency enforcement. Considered termination conditions are based on suitable lower and upper bounds for polynomial functions over boxes that are determined using new results proved in the paper. The algorithm is particularly appropriate to reason on high-degree polynomial constraints because the proposed method to determine lower and upper bounds can outperform alternative methods when high-degree polynomials in a moderate number of variables are considered.

中文翻译：

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对有限域上的高次多项式约束进行推理的细分算法

本文提出了一种算法来推理以具有整数系数的多项式表示的约束，其变量取整数的有限子集中的值。所提出的算法假设变量域的初始近似值就边界框而言是可用的，并且它递归地将框细分为不相交的框，直到满足终止条件。该算法包括三个终止条件，允许将其用于三个相关的推理任务：约束满足、解决方案枚举和超弧一致性执行。考虑的终止条件基于框上多项式函数的合适下限和上限，这些框是使用论文中证明的新结果确定的。