By Matthias Beck, Sinai Robins
This textbook illuminates the sphere of discrete arithmetic with examples, thought, and purposes of the discrete quantity of a polytope. The authors have weaved a unifying thread via easy but deep principles in discrete geometry, combinatorics, and quantity conception.
We come across the following a pleasant invitation to the sphere of "counting integer issues in polytopes", and its a variety of connections to straightforward finite Fourier research, producing services, the Frobenius coin-exchange challenge, reliable angles, magic squares, Dedekind sums, computational geometry, and extra.
With 250 routines and open difficulties, the reader seems like an energetic player.
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Extra info for Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra (Undergraduate Texts in Mathematics)
Sample text
There are several special cases of A = {a1 , a2 , . . , ad } for which the Frobenius problem is solved, for example, arithmetic sequences [153, Chapter 3]. 34. 41. , in light of the Morales–Denham theorem mentioned in the Notes. , arithmetic sequences. 42. For which 0 ≤ n ≤ b − 1 is sn (a1 , a2 , . . , ad ; b) = 0? 2 A Gallery of Discrete Volumes Few things are harder to put up with than a good example. Mark Twain (1835–1910) A unifying theme of this book is the study of the number of integer points in polytopes, where the polytopes lives in a real Euclidean space Rd .
25), we’re counting integer vectors n ∈ Zd≥0 satisfying An − tb = 0 , that is, An = tb . 13 (Euler’s generating function). 23). Then the Ehrhart quasipolynomial of P can be computed as follows: LP (t) = const 1 (1 − zc1 ) (1 − zc2 ) · · · (1 − zcd ) ztb . ⊓ ⊔ We finish this section with rephrasing this constant-term identity in terms of Ehrhart series. 14. 23). Then the Ehrhart series of P can be computed as 1 (1 − zc1 ) (1 − zc2 ) · · · (1 − zcd ) 1 − EhrP (x) = const . x zb Proof. 13, EhrP (x) = const t≥0 1 (1 − zc1 ) (1 − zc2 ) · · · (1 − zcd ) ztb 1 = const c 1 (1 − z ) (1 − zc2 ) · · · (1 − zcd ) = const t≥0 xt xt ztb 1 1 (1 − zc1 ) (1 − zc2 ) · · · (1 − zcd ) 1 − zxb .
24. ⊓ ⊔ Pick’s theorem allows us not only to count the lattice points strictly inside the polygon P but also the total number of lattice points contained in P, because this number is 1 1 I + B = A − B + 1 + B = A + B + 1. 9. Suppose P is an integral convex polygon with area A and B lattice points on its boundary. (a) The lattice-point enumerator of P is the polynomial 1 LP (t) = A t2 + B t + 1 . 2 (b) Its evaluation at negative integers yields the relation LP (−t) = LP ◦ (t) . (c) The Ehrhart series of P is A− EhrP (z) = B 2 + 1 z2 + A + (1 − z)3 B 2 −2 z+1 .