Sinai Robins

The Coin-Exchange Problem of Frobenius

Suppose we are interested in an infinite sequence of numbers \(\left (a_{k}\right )_{k=0}^{\infty }\) that arises naturally from geometric problems, or from recursively defined problems. Is there a “good formula” for a k as a function of k? Are there identities involving various a k ’s? Embedding this sequence into the generating function \(F(z) =\sum _{k\geq 0}a_{k}\,z^{k}\) allows us to retrieve answers to the questions above in a surprisingly quick and elegant way. We may think of F(z) as lifting our sequence a k from its discrete setting into the continuous world of functions. We introduce techniques for working with generating functions, and we use them to shed light on the Frobenius coin-exchange problem: Given relatively prime positive integers \(a_{1},a_{2},\ldots,a_{n}\), what is the largest integer that cannot be written as a nonnegative integral linear combination of \(a_{1},a_{2}\ldots,a_{n}\)?

Finite Fourier Analysis

We now consider the vector space of all complex-valued periodic functions on the integers with period b. It turns out that every such function a(n) on the integers can be written as a polynomial in the bth root of unity \(\xi ^{n}:= e^{2\pi in/b}\). Such a representation for a(n) is called a finite Fourier series. Here we develop finite Fourier theory using rational functions and their partial fraction decomposition. We then define the Fourier transform and the convolution of finite Fourier series, and show how one can use these ideas to prove identities on trigonometric functions, as well as find connections to the classical Dedekind sums.

A Discrete Version of Green’s Theorem Using Elliptic Functions

We now allow ourselves the luxury of using basic complex analysis. In particular, we assume in this chapter that the reader is familiar with contour integration and the residue theorem. We may view the residue theorem as yet another result that intimately connects the continuous and the discrete: it transforms a continuous integral into a discrete sum of residues. Using the Weierstras \(\wp \)- and ζ-functions, we show here that Pick’s theorem is a discrete version of Green’s theorem in the plane. As a bonus, we also obtain an integral formula for the discrepancy between the area enclosed by a general curve C and the number of integer points contained in C.

Dedekind Sums, the Building Blocks of Lattice-Point Enumeration

We encountered Dedekind sums in our study of finite Fourier analysis in Chapter 7, and we became intimately acquainted with their siblings in our study of the coin-exchange problem in Chapter 1 They have one shortcoming, however (which we shall remove): the definition of s(a, b) requires us to sum over b terms, which is rather slow when b = 2100, for example. Luckily, there is a magical reciprocity law for the Dedekind sum s(a, b) that allows us to compute it in roughly \(\log _{2}(b) = 100\) steps in this example. This is the kind of magic that saves the day when we try to enumerate lattice points in integral polytopes of dimension d ≤ 4. In this chapter, we focus on the computational-complexity issues that arise when we try to compute Dedekind sums explicitly. In many ways, the Dedekind sums extend the notion of the greatest common divisor of two integers.

Euler–Maclaurin Summation in ℝd

Thus far, we have often been concerned with the difference between the discrete volume of a polytope \(\mathcal{P}\) and its continuous volume. An important extension is the difference between the discrete integer-point transform and its continuous sibling:

The Decomposition of a Polytope into Its Cones

\displaystyle{ \sum _{\mathbf{m}\in \mathcal{P}\cap \mathbb{Z}^{d}}\!\!e^{\mathbf{m}\cdot \mathbf{x}} -\int _{ \mathcal{P}}e^{\mathbf{y}\cdot \mathbf{x}}d\mathbf{y}\,, }

Counting Lattice Points in Polytopes: The Ehrhart Theory

(12.1) where we have replaced the variable \(\mathbf{z}\) that we have commonly used in generating functions by an exponential variable. Note that on setting x = 0 in (12.2), we obtain the difference between the discrete and the continuous volumes of \(\mathcal{P}\). Relations between the two quantities \(\sum _{\mathbf{m}\in \mathcal{P}\cap \mathbb{Z}^{d}}e^{\mathbf{m}\cdot \mathbf{x}}\) and \(\int _{\mathcal{P}}e^{\mathbf{y}\cdot \mathbf{x}}d\mathbf{y}\) are known as Euler–Maclaurin summation formulas for polytopes. The “behind-the-scenes” operators that are responsible for affording us with such connections are the differential operators known as Todd operators, whose definition utilizes the Bernoulli numbers in a surprising way.

A Gallery of Discrete Volumes

Our first goal in this chapter is to introduce and prove a set of fascinating identities, known as the Dehn–Sommerville relations, which are linear relations among the face numbers of a polytope. Our second goal is to unify the Dehn–Sommerville relations with Ehrhart–Macdonald reciprocity.