论文标题
多项式多项式：计算问题的工具箱
A Plethora of Polynomials: A Toolbox for Counting Problems
论文作者
论文摘要
组合设备和离散优化的各种问题取决于计算多元层中整数点的集合$ s $，或者是通过离散几何和一阶逻辑构建的一些更通用的对象。我们参观了许多此类问题。特别是，我们认为此类集合的系列$ s_t $，具体取决于一个或多个整数参数$ t $，并分析函数$ f（t）= | s_t | $的行为。在我们研究的示例中，此功能表现出令人惊讶的多项式行为。我们以两个广泛的定理详细介绍了这种多项式行为必须保持。大量示例说明了这种行为发生的框架，并为许多证据提供了直觉，从而帮助我们创建了一个用于计算此类问题的工具箱。
A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour through numerous problems of this type. In particular, we consider families of such sets $S_t$ depending on one or more integer parameters $t$, and analyze the behavior of the function $f(t)=|S_t|$. In the examples that we investigate, this function exhibits surprising polynomial-like behavior. We end with two broad theorems detailing settings where this polynomial-like behavior must hold. The plethora of examples illustrates the framework in which this behavior occurs and also gives an intuition for many of the proofs, helping us create a toolbox for counting problems like these.
