A Finite Calculus Approach to Ehrhart Polynomials. Kevin Woods, Oberlin College (joint work with Steven Sam, MIT)

Transcription

1 A Finite Calculus Approach to Ehrhart Polynomials Kevin Woods, Oberlin College (joint work with Steven Sam, MIT)

2 Ehrhart Theory Let P R d be a rational polytope L P (t) = #tp Z d Ehrhart s Theorem: L p (t) = c d (t)t d + c d 1 (t)t d c 0 (t), where c i (t) are periodic. When P is integral, period = 1, so L p (t) is a polynomial.

3 An Analogy L P (t) is the discrete analog of volume 1 = L P (t). a tp Z d 1 dx = vol(tp) = vol(p)t d. tp. L P (t) vol(p)t d

4 Push the Analogy Let d be the convex hull of the origin the standard basis vectors e i. Compute the volume of t d

5 Push the Analogy Let d be the convex hull of the origin the standard basis vectors e i. Compute the volume of t d vol(t d ) = Inductively, t 0 vol(s d 1 ) ds vol(t d ) = td d!.

6 Push the Analogy Why it works so nice: t 0 t 0 is a linear operator. acts nicely on a basis of R[x]. t 0 s n = 1 n + 1 tn+1.

7 Push the Analogy Discrete version: L d (t) = Inductively, t L d 1 (s). L d (t) = (t + 1)(t + 2) (t + d). d!

8 Push the Analogy Why it works so nice: t is a linear operator. t acts nicely on a basis of R[x]. t s n = 1 n + 1 (t + 1)n+1, where t n = t(t 1)(t 2) (t d + 1).

9 Is it always this easy?

10 Is it always this easy? No.

11 A Harder One L P (t) =?

12 A Harder One L P (t) =?

13 A Harder One L Q (s) = s + 2 s s + 2 = 2 2 2

14 A Harder One s L Q (s) = + 2 s s + 2 = t L P (t) = L Q (s) 2t s + 2 = 2 2 = a polynomial!!??!!

15 A Harder One 2t s + 2 L P (t) = 2 2 t ( ) (2r 1) + 2 2r + 2 = = 1 + r=1 t 4r r=1 = 4r 2 + 4r + 1

16 A Harder One 2t s + 2 L P (t) = 2 2 t ( ) (2r 1) + 2 2r + 2 = = 1 + r=1 t 4r r=1 = 4r 2 + 4r + 1

17 A Harder One 2t s + 2 L P (t) = 2 2 t ( ) (2r 1) + 2 2r + 2 = = 1 + r=1 t 4r r=1 = 4r 2 + 4r + 1

18 Is it always this easy?

19 Is it always this easy? Yes.

20 Tools Quasi-polynomial version of finite calculus: If f (s) is a quasi-polynomial with period r, then at b F (t) = f (s) is a quasi-polynomial with period rb gcd(a, r). Key: It s the smallest t such that at b is integer multiple of r.

21 Tools Triangulation. Summation only works for pyramids. Induction. Need quasi-polynomial version even to get polynomial version. Bug? Feature?

22 Periodicity! For Free! Careful triangulation/induction immediately gives us: Theorem (McMullen) Given Ehrhart quasi-polynomial L P (t) = c 0 (t) + c 1 (t)t + + c d (t)t d, and given r and i such that the affine hull of rf contains integer points, for all i-dimensional faces F. Then r is a period of c i (t).

23 Periodicity! For Free! Let D be smallest positive integer such that DP is integral. Then D is a period of each c i (t). If P is integral, D = 1 and L P (t) is a polynomial. If P is full-dimensional: Affine hull of 1 P contains integer points. Period of c d (t) is 1. (c d (t) = vol(p))

24 Reciprocity! For Free! Theorem (Ehrhart-Macdonald Reciprocity) If P is the relative interior of P and L P (t) = #tp Z d, L P (t) = ( 1) dim(p) L P ( t).

25 Reciprocity! For Free! Reciprocity in Finite Calculus If f (t) a (quasi-)polynomial, is a (quasi-)polynomial, F (n). n f (s) n f (s)

26 Reciprocity! For Free! Reciprocity in Finite Calculus If f (t) a (quasi-)polynomial, is a (quasi-)polynomial, F (n). n f (s) n f (s) = 1 s= n+1 f (s)

27 Reciprocity! For Free! Reciprocity in Finite Calculus If f (t) a (quasi-)polynomial, is a (quasi-)polynomial, F (n). n f (s) n f (s) = 1 s= n+1 = F ( n) f (s)

28 Reciprocity! For Free! t 1 L P (t) = L Q (s) s=1 t 1 = ±L Q ( s) s=1 t = ±L Q (s) = ± L P ( t)

29 Reciprocity! For Free! t 1 L P (t) = L Q (s) s=1 t 1 = ±L Q ( s) s=1 t = ±L Q (s) = ± L P ( t)

30 Reciprocity! For Free! t 1 L P (t) = L Q (s) s=1 t 1 = ±L Q ( s) s=1 t = ±L Q (s) = ± L P ( t)

31 Reciprocity! For Free! t 1 L P (t) = L Q (s) s=1 t 1 = ±L Q ( s) s=1 t = ±L Q (s) = ± L P ( t)

32 Okay, Not Quite Free You do have to be careful with the triangulation and Inclusion-Exclusion for reciprocity. Need Euler characteristic, topologically or combinatorially. True of most proofs (though check out irrational version, Beck Sottile).

33 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

34 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

35 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

36 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

37 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

38 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

39 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

40 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

41 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

42 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

43 Context Classic Proofs Ehrhart, Stanley (generating functions) McMullen (valuations) Contemporary (simpler) Proofs Beck (partial fractions) Sam (full-dimensional Inclusion-Exclusion)

44 Questions Should we pick a nice basis to write (quasi-)polynomial? 1, t + 1, (t + 1)(t + 2)/2,...

45 Questions Should we pick a nice basis to write (quasi-)polynomial? 1, ( t + 1, (t + 1)(t + 2)/2,... ) ( t+d d, t+d 1 ) ( d,..., t ) d (for polynomials of degree at most d)

46 Questions Should we pick a nice basis to write (quasi-)polynomial? 1, ( t + 1, (t + 1)(t + 2)/2,... ) ( t+d d, t+d 1 ) ( d,..., t ) d (for polynomials of degree at most d) If then L P (t) = d j=0 ( ) t + d j h j, d L P (s)t s = h 0 + h 1 t + + h d t d (1 t) d+1.

47 Questions Does this translate into an algorithm? t 2s + 3 =? 4

48 Questions Does this translate into an algorithm? t 2s + 3 =? 4 t 2s + 3 3s + 2 =? 4 5

49 Questions Does this translate into an algorithm? Best I can say: t 2s + 3 =? 4 t 2s + 3 3s + 2 =? 4 5 Translate to/from generating functions (Verdoolaege W). Apply Barvinok s algorithm.

50 Thank You!