AI 中文总结
研究次数为\(d\)且周期整除\(p\)的最终非递减拟多项式，通过刻画此类拟多项式，详细分析固定次数和周期且\(h\)向量非负的拟多项式空间，确定其数量增长速率与第\(0\)个组成多项式\(h\)向量元素之和的函数关系。
AI 中文摘要
拟多项式在组合数学和代数中无处不在，因为它们出现在各种枚举问题中。由于拟多项式由组成多项式构成，其行为比单个多项式更为微妙。特别是，与多项式不同，定义在正整数上的拟多项式可能有无穷多个递减点。在这项工作中，我们刻画了次数为\(d\)且周期整除\(p\)的最终非递减拟多项式，即只有有限多个值递减的拟多项式。然后，我们对固定次数\(d\)且周期整除固定\(p\)且拟多项式的\(h\)向量非负的最终非递减拟多项式空间进行了详细分析。利用这一分析，我们确定了此类拟多项式数量的增长速率，它是关于第\(0\)个组成多项式的\(h\)向量元素之和的函数。
英文摘要
Quasi-polynomials are ubiquitous in combinatorics and algebra, as they arise in a variety of enumeration problems. Because quasi-polynomials consist of constituent polynomials, their behavior is more subtle than for a single polynomial. In particular, unlike for a polynomial, it is possible for a quasi-polynomial defined on the positive integers to have infinitely many points at which it is decreasing. In this work, we characterize quasi-polynomials of degree $d$ and period dividing $p$ that are eventually nondecreasing, i.e., that have only finitely many values at which they decrease. We then give a detailed analysis of the space of eventually nondecreasing quasi-polynomials with fixed degree $d$ and period dividing a fixed $p$ such that the $h$-vector of the quasi-polynomial is nonnegative. Using this analysis, we determine the rate of growth of the number of such quasi-polynomials as a function of the sum of the $h$-vector entries for the $0$-th constituent polynomial.
