The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums
The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums
复制标题
弗罗贝尼乌斯问题、有理多面体和傅立叶-戴德金和
DOI：
--
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
S. Robins
中科院分区：
文献类型：
--
作者：
M. Beck;Ricardo Diaz;S. Robins
研究了有理多胞形P={（x1，.，xn）∈R <$0 n：∑k= 1 nxkak <$1}的整数扩张中的格点数，其中a1，.，an为正整数.这个多面体与弗罗贝纽斯的线性丢番图问题密切相关：给定互质的正整数a1，...，an，找到t（弗罗贝纽斯数）的最大值，使得m1 a1+···+mnan=t在正整数m1，...，mn中没有解。这等价于找到最大的膨胀tP使得小平面{∑k= 1 nxkak =t}不包含晶格点的问题。给出了Ehrhart拟多项式L（P，t）<$#（tP <$Zn）和L（P°，t）<$#（tP° <$Zn）的两种计算方法.在计算中，出现了类似戴德金的有限傅立叶和。我们得到了这些款项的倒易律，推广了Gessel定理。作为我们的公式的推论，我们重新推导Zagier的高维Dedekind和的互易定律。最后，我们找到了Fourier-Dedekind和的界，并利用它们给出了Frobenius数的新界。
We study the number of lattice points in integer dilates of the rational polytope P={(x1,…,xn)∈R⩾0n:∑k=1nxkak⩽1}, where a1,…,an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,…,an, find the largest value of t (the Frobenius number) such that m1a1+···+mnan=t has no solution in positive integers m1,…,mn. This is equivalent to the problem of finding the largest dilate tP such that the facet {∑k=1nxkak=t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L(P,t)≔#(tP∩Zn) and L(P°,t)≔#(tP°∩Zn). Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier–Dedekind sums and use them to give new bounds for the Frobenius number.