The degree of lattice polytopes

Transcription

1 The degree of lattice polytopes Benjamin Nill - FU Berlin Graduiertenkolleg MDS - December 10, 2007 Lattice polytopes having h -polynomials with given degree and linear coefficient. arxiv: , to appear in Eur. J. Comb.

2 1.

3 Lattice polytopes Throughout: M := Z r lattice, M R := R r vector space. P M R is lattice polytope (i.e., convex hull of finitely many lattice points). P has dimension n (n = dimension of smallest affine space P ). Yes: No:

4 Isomorphism classes Definition: Lattice polytopes P, P are isomorphic, if there is a matrix G GL n (Z) and b M such that the affine lattice automorphism M M : m G m + b maps the vertices of P onto the vertices of P. Interested in invariants under these isomorphisms (e.g., number of lattice points, volume).

5 The Ehrhart polynomial Theorem. [Ehrhart 1962] There is a polynomial, called Ehrhart polynomial, L P (x) Q[x] of degree n such that L P (k) = kp M k N. Example: P 2P 3P L (1) = 3 L (2)=6 L (3) = 10 P P P L P (x) = 1 2 x2 + 3 ( ) x x + 1 =. 2

6 Let us write L P (x) = n i=0 h i for coefficients h 0,..., h n Q. ( x + n i n ),

7 Let us write L P (x) = n i=0 h i for coefficients h 0,..., h n Q. One can compute ( ) k + n i t k = n k N ( x + n i n ), t i (1 t) n+1.

8 Let us write L P (x) = n i=0 h i for coefficients h 0,..., h n Q. One can compute ( ) k + n i t k = n This implies k N kp M t k = k N ( x + n i n ), t i (1 t) n+1. h P(t) (1 t) n+1, where h P(t) = n i=0 h i t i is called h -polynomial.

9 Theorem. [Stanley 1980] The coefficients h 0,..., h n are natural numbers. Observations: h 0 = 1. h 1 = P M (n + 1). h n = int(p ) M. h h n = n! vol(p ) =: Vol(P ) called normalized volume.

10 Example: n = 2 h P(t) = 1 + ( P M 3) t + int(p ) M t t+8t² Pick s formula: Vol(P ) = (P ) M + 2 int(p ) M 2, vol(p ) = (P ) M /2 + int(p ) M 1.

11 2.

12 The degree Definition: The degree of P is defined as the degree of the h -polynomial. Hence deg(p ) := max(i : h i 0). 0 deg(p ) n = dim(p ). Philosophy: The degree of a lattice polytope may be regarded as its true dimension.

13 The codegree Definition: The codegree of P is defined as: codeg(p ) := min{k Z 0 : int(kp ) M } Reciprocity-Theorem 1 codeg(p ) = n + 1 deg(p ) n + 1, h deg(p ) = int(codeg(p )P ) M.

14 Example: n = 2 P 2P 3P 4P codeg 3 codeg 2 codeg 1 codeg t 1+7t+t² 1+12t+3t² Observations: codeg(p ) = 1 deg(p ) = n int(p ) M codeg(p ) = n + 1 deg(p ) = 0 P is unimodular simplex.

15 3. Classification of lattice polytopes of degree one

16 Dimension two Let P M R be a lattice polygon. deg(p ) 1 P has no interior lattice points. Theorem. [Arkinstall 1980; Koelman 1991; Khovanskii 1997; Schicho 2003] There are precisely two cases, in which P has no interior lattice points:

17 either P is the exceptional triangle S: S 1+3t or P lies between two parallel hyperplanes of integral distance one: 1 1+t 1+t 1+2t

18 Lattice pyramid construction Definition: The lattice pyramid over P is defined as Pyr(P ) := conv({0}, P {1}) M R R. P Pyr(P) lattice distance one Recursively: k-fold pyramid over P : Pyr k (P ) := Pyr(Pyr k 1 (P )). Proposition: P and Pyr(P ) have same h -polynomial. Degree and normalized volume is unchanged!

19 Higher dimensions Definition: An n-dimensional exceptional simplex is an (n 2)-fold lattice pyramid over S. Example: n = 3 Pyr(S) 1+3t

20 Definition: An n-dimensional Lawrence prism with heights h 1,..., h n : h = 2 1 h = 1 2 e n h = 1 n e 0 e n 1 e t 1+3t 1+5t

21 Classification result Theorem. [Batyrev, N. 2006] A lattice polytope has degree 1 if and only if it is an exceptional simplex or a Lawrence prism. Proof by induction on the number of lattice points, using the monotonicity property of the degree: Theorem. [Stanley 1993] Q P lattice polytopes h Q h P coefficientwise.

22 4. How many lattice polytopes with given h -polynomial?

23 Case of degree one Example: h = 1 + t (deg = 1 and Vol = 2): n = 1: P 1 n = 2: Pyr(P) 1 P 2 n 3: n 1 n 2 P 1 P2 Pyr ( ) Pyr ( ) Proposition: An n-dimensional lattice polytope P of degree one is a lattice pyramid, if n > Vol(P ).

24 The general case Theorem. [Batyrev 2006] A lattice polytope of degree d, normalized volume V, and dimension n is a lattice pyramid, if ( ) 2d + V 1 n 4d. 2d Proof relies heavily on commutative algebra.

25 Definition: Let d, V be fixed. C(n) := number of isomorphism classes of n-dimensional lattice polytopes of degree d and normalized volume V. Then C(n) is finite (theorem of Lagarias and Ziegler), upper monotone in n (lattice pyramid construction), becomes constant for n large (Batyrev s result). Corollary: There are only finitely many lattice polytopes with the same h -polynomial up to isomorphisms and lattice pyramid constructions.

26 5. What about lattice polytopes of given degree?

27 The Cayley polytope conjecture Definition: A lattice polytope P is a Cayley polytope P = P 1 P 2, if P P 2 lattice distance one P 1 On the other hand, let P 1,..., P s M R be given. Definition: P 1 P s is defined as conv(p 1 {e 1 },..., P s {e s }) M R Z s, where e 1,..., e s is a lattice basis of Z s.

28 Examples: Lattice pyramids are Cayley polytopes. Lawrence prisms are Cayley polytopes of segments. [0,2]*[0,3]*[0,1]

29 Examples: Lattice pyramids are Cayley polytopes. Lawrence prisms are Cayley polytopes of segments. [0,2]*[0,3]*[0,1] Conjecture [Batyrev, N.]: Let d be fixed. There exists N such that any lattice polytope of degree d and dimension n is a Cayley polytope, if n N. Example (d d = 1): Conjecture holds with N = 3.

30 A surprise... Theorem I. [N. 2007] A lattice simplex of degree d and dimension n is a lattice pyramid, if n 4d 1.

31 A surprise... Theorem I. [N. 2007] A lattice simplex of degree d and dimension n is a lattice pyramid, if n 4d 1. Example (d d = 1): Exceptional simplices are lattice pyramids for n 3. Lawrence prisms that are simplices are lattice pyramids for n 2.

32 ... and its generalization Theorem II. [N. 2007] A lattice polytope of degree d and dimension n is a lattice pyramid, if n (f 0 n 1)(2d + 1) + 4d 1, where f 0 equals the number of vertices of P. Proofs of Theorems I and II are purely combinatorial.

33 Back to the h -polynomial f 0 n 1 P M n 1 = h 1 Corollary: A lattice polytope of degree d and dimension n is a lattice pyramid, if n h 1(2d + 1) + 4d 1.

34 Back to the h -polynomial f 0 n 1 P M n 1 = h 1 Corollary: A lattice polytope of degree d and dimension n is a lattice pyramid, if n h 1(2d + 1) + 4d 1. Fine tuning of Theorem II Batyrev s result revised: Corollary: A lattice polytope of degree d, normalized volume V, and dimension n is a lattice pyramid, if n (V 1)(2d + 1).

35 6. Theorem I. A lattice simplex of degree d and dimension n is a lattice pyramid, if n 4d 1.

36 The setup Let P be an n-dimensional lattice simplex of degree d. P := conv(v 0,..., v n ) M R {1}, M := M Z. Definition: (half-open) parallelepiped n Π(P ) := { λ i v i : 0 λ i < 1}. i=0 v 0 + v1 + v2 v v v v v 0 P v 2 v v v 1

37 A direct way to read off h P h i = {m Π(P ) M : ht(m) = i}, where ht(m) equals the last coordinate of m. Example: P = S the exceptional triangle, h = 1 + 3t. ht=3 v 0 + v1 + v2 ht=2 v v v v ht=1 v 0 v 2 v v v 1 ht=0

38 The support 1. Let m M R R. Then n m = λ i v i for λ i R, i=0 supp(m) := {i {0,..., n} : λ i 0}. 2. supp(p ) := supp(m). m Π(P ) M

39 A simple reformulation Lemma: P lattice pyramid supp(p ) {0,..., n}

40 A simple reformulation Lemma: P lattice pyramid supp(p ) {0,..., n} Proof: Follows from Vol(P ) = h h n = Π(P ) M.

41 A simple reformulation Lemma: P lattice pyramid supp(p ) {0,..., n} Proof: Follows from Vol(P ) = h h n = Π(P ) M. Let P := conv(v 0,..., v n 1 ). Then t.f.a.e.: P is a lattice pyramid over P (with apex v n ) Vol(P ) = Vol(P ) Π(P ) M = Π(P ) M supp(p ) {0,..., n 1}

42 Line of argument Goal: supp(p ) 4d 1.

43 Line of argument Goal: supp(p ) 4d 1. Idea: Cover the support of P in a greedy manner. Choose recursively lattice points m 0, m 1, m 2,... Π(P ) M such that supp(m 0 ) maximal, supp(m 0 ) supp(m 1 ) maximal, supp(m 0 ) supp(m 1 ) supp(m 2 ) maximal,...

44 Claim: supp(m 0 ) 1 2d, supp(m 0 ) supp(m 1 ) ( ) 2d, supp(m 0 ) supp(m 1 ) supp(m 2 ) ( ) 2d,...

45 Claim: supp(m 0 ) 1 2d, supp(m 0 ) supp(m 1 ) ( ) 2d, supp(m 0 ) supp(m 1 ) supp(m 2 ) ( ) 2d,... Finiteness yields supp(p ) = supp(m j ) < 2 2d = 4d. j=0 Therefore Proof of claim by induction. supp(p ) 4d 1.

46 Proof of claim for m 0 We have to show m Π(P ) M supp(m) 2d. Since m Π(P ) M, d = deg(p ) ht(m). Let supp(m) = {0,..., s}, P := conv(v 0,..., v s ) m int(π(p )) M.

47 Proof of claim for m 0 We have to show m Π(P ) M supp(m) 2d. Since m Π(P ) M, d = deg(p ) ht(m). Let supp(m) = {0,..., s}, P := conv(v 0,..., v s ) m int(π(p )) M. This yields d ht(m) codeg(p ) = dim(p ) + 1 deg(p ) Hence s + 1 deg(p ) = s + 1 d. supp(m) = s + 1 2d.

48 Proof of induction step m 0 m 1 Schematic figure of supp(m 0 ) supp(m 1 ) {0,..., n}: m 0 m 1 a b c b + c = supp(m 1 ) supp(m 0 ) = a + b. (1) It suffices to show c d = deg(p ). Since then supp(m 0 ) supp(m 1 ) = a + b + c 2d + d = ( )2d.

49 n n m 0 = λ i v i, m 1 = µ i v i. i=0 i=0 Then n m 1 := {λ i + µ i }v i Π(P ) M, i=0 where {γ} [0, 1[ is the fractional part of a real number γ.

50 n n m 0 = λ i v i, m 1 = µ i v i. i=0 i=0 Then n m 1 := {λ i + µ i }v i Π(P ) M, i=0 where {γ} [0, 1[ is the fractional part of a real number γ. m 0 m 1 m 1 a b c so a + c supp(m 1) supp(m 0 ) = a + b, c b. (2)

51 Combining and yields so b + c a + b (1) c b (2) c b a + b c, 2c a + b = supp(m 0 ) 2d, c d. This finishes the proof of the induction step m 0 m 1. The general induction step works similarly.

52 Hence, we proved Theorem I: Theorem I. A lattice simplex of degree d and dimension n is a lattice pyramid, if n 4d 1. However general Cayley conjecture still open...

53 7.

54 The leading coefficient conjecture Conjecture: Let i {1,..., d}. There exists a constant N such that any n-dimensional lattice polytope of degree d and with coefficient h i is a lattice pyramid, if n N. For i = d equivalent to: LC-Conjecture [Batyrev]: There exists a uniform upper bound on the normalized volume of lattice polytopes of degree d and leading coefficient h d. Holds for d 2 due to Treutlein (2007).

55 The Cayley conjecture - refined RC-Conjecture: Let d be fixed. There exists N such that any lattice polytope P of degree d and dimension n N is a Cayley polytope P 1 P s of nonempty lattice polytopes for s = n + 2 N. Holds for d = 1 with N = 3. Theorem. [Haase, N., Payne 2007] RC-Conjecture implies LC-Conjecture.

56 Gorenstein polytopes Definition: P is Gorenstein, if h P is symmetric. In particular, h d = h 0 = 1. Proposition [N. 2007]: RC-Conjecture holds for Gorenstein polytopes (with N = 2d). Corollary: There exists a uniform bound on the normalized volume of Gorenstein polytopes of degree d. Proposition follows from combining [Batyrev, Borisov 1997] criterion for Gorenstein polytopes in terms of Gorenstein cones, [Batyrev, N. 2007] characterization of Cayley polytopes by special simplices.