By Matthias Beck, Sinai Robins
This textbook illuminates the sector of discrete arithmetic with examples, idea, and purposes of the discrete quantity of a polytope. The authors have weaved a unifying thread via easy but deep rules in discrete geometry, combinatorics, and quantity idea.
We stumble upon the following a pleasant invitation to the sphere of "counting integer issues in polytopes", and its a number of connections to effortless finite Fourier research, producing capabilities, the Frobenius coin-exchange challenge, sturdy angles, magic squares, Dedekind sums, computational geometry, and extra.
With 250 routines and open difficulties, the reader appears like an energetic player.
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Extra info for Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra (Undergraduate Texts in Mathematics)
I) ♣ If m ∈ Z>0 , n ∈ Z, then n−1 = − −n m m − 1. n−1 (j) ♣ If m ∈ Z>0 , n ∈ Z, then m + 1 is the least integer greater than or equal to n/m. 5. Rewrite in terms of the fractional-part function as many of the above identities as you can make sense of. 6. Suppose m and n are relatively prime positive integers. Prove that m−1 kn = m k=0 n−1 j=0 1 jm = (m − 1)(n − 1) . 7. Prove the following identities. They will become handy at least twice: when we study partial fractions, and when we discuss finite Fourier series.
Let ∆ be the standard d-simplex. (a) The lattice-point enumerator of ∆ is the polynomial L∆ (t) = d+t d . (b) Its evaluation at negative integers yields (−1)d L∆ (−t) = L∆◦ (t). 1 (c) The Ehrhart series of ∆ is Ehr∆ (z) = (1−z) d+1 . 4 The Bernoulli Polynomials as Lattice-Point Enumerators of Pyramids There is a fascinating connection between the Bernoulli polynomials and certain pyramids over unit cubes. The Bernoulli polynomials Bk (x) are defined through the generating function z exz = ez − 1 k≥0 Bk (x) k z k!
It turns out to be 7 2 5 7 + (−1)t t + t+ . 23), we denote the columns of A by c1 , c2 , . . , cd . Let z = (z1 , z2 , . . 25) c c 2 1 (1 − z ) (1 − z ) · · · (1 − zcd ) ztb in terms of geometric series: n1 ≥0 zn 1 c 1 n2 ≥0 zn 2 c2 · · · nd ≥0 zn d c d 1 . ztb cm Here we use the abbreviating notation zc := z1c1 z2c2 · · · zm for the vectors m m z = (z1 , z2 , . . , zm ) ∈ C and c = (c1 , c2 , . . , cm ) ∈ Z . When multiplying out everything, a typical term will look like n1 c1 + n2 c2 + · · · + nd cd − tb = An − tb , where n = (n1 , n2 , .
Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra (Undergraduate Texts in Mathematics) by Matthias Beck, Sinai Robins