Minkowski Length of Lattice Polytopes

Transcription

1 Minkowski Length of Lattice Polytopes Einstein Workshop on Lattice Polytopes Ivan Soprunov (with Jenya Soprunova) Cleveland State University December 12, 2016 Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 1/15

2 Counting rational points on hypersurfaces Tsfasman s Question (1989, Luminy): What is the largest number of F q -points on a hypersurface H P d of degree t? Thatis, N q (t, d) = max deg f =t {p 2 Pd (F q ) f (p) =0}, over all homogeneous f 2 F q [x 0,...,x d ] of degree t? Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 2/15

3 Counting rational points on hypersurfaces Tsfasman s Question (1989, Luminy): What is the largest number of F q -points on a hypersurface H P d of degree t? Thatis, N q (t, d) = max deg f =t {p 2 Pd (F q ) f (p) =0}, over all homogeneous f 2 F q [x 0,...,x d ] of degree t? Serre s Answer (1989): Forq t, the polynomials that factor the most have the most zeroes over F q. Thus we should take f to be a product of linear factors and so N q (t, d) =t qd 1 q 1 (t 1) qd 1 1 q 1 = tq d 1 + q d Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 2/15

4 Counting rational points on hypersurfaces More generally, let X be a toric variety and D is a T-invariant Cartier divisor on X with polytope P = P D.ThentheF q -global sections of O X (D) canbeidentifiedwith L(P) := M F q x a. a2p\z d Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 3/15

5 Counting rational points on hypersurfaces More generally, let X be a toric variety and D is a T-invariant Cartier divisor on X with polytope P = P D.ThentheF q -global sections of O X (D) canbeidentifiedwith L(P) := M F q x a. a2p\z d Question: Given P, what is the largest number of zeroes N q (P) in (F q) d a polynomial f 2 L(P) mayhave? Example: N q (t )=t(q 1) d 1, by Serre s argument Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 3/15

6 Counting rational points on hypersurfaces More generally, let X be a toric variety and D is a T-invariant Cartier divisor on X with polytope P = P D.ThentheF q -global sections of O X (D) canbeidentifiedwith L(P) := M F q x a. a2p\z d Question: Given P, what is the largest number of zeroes N q (P) in (F q) d a polynomial f 2 L(P) mayhave? Example: N q (t )=t(q 1) d 1, by Serre s argument Easier Questions: I What is the largest number of factors a polynomial f 2 L(P) may have? I What do the factors in this case look like? Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 3/15

7 Counting rational points on hypersurfaces In fact, the easier questions are about the geometry of P. I The largest number of factors f 2 L(P) mayhaveisthe Minkowski length of P. I The irreducible factors of such f have Newton polytopes that are strongly indecomposable. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 4/15

8 Minkowski length: Definition Let P be a lattice polytope in R d. Definition: The largest number of lattice polytopes of positive dimension whose Minkowski sum is contained in P is called the Minkowski length: L(P) =max{l 2 N Q = Q Q L P, dim Q i > 0}. Lattice polytopes with L(P) = 1 are strongly indecomposable. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 5/15

9 Minkowski length: Definition Let P be a lattice polytope in R d. Definition: The largest number of lattice polytopes of positive dimension whose Minkowski sum is contained in P is called the Minkowski length: L(P) =max{l 2 N Q = Q Q L P, dim Q i > 0}. Lattice polytopes with L(P) = 1 are strongly indecomposable. Example I L(P) =3 I L(t )=t for a unimodular simplex and t 2 N. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 5/15

10 Minkowski length: Definition Let P be a lattice polytope in R d. Definition: The largest number of lattice polytopes of positive dimension whose Minkowski sum is contained in P is called the Minkowski length: L(P) =max{l 2 N Q = Q Q L P, dim Q i > 0}. Lattice polytopes with L(P) = 1 are strongly indecomposable. Example I L(P) =3 I L(t )=t for a unimodular simplex and t 2 N. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 5/15

11 Minkowski length: Properties Simple Properties: I Invariance: L(P) isagl(d, Z)-invariant, I Monotonicity: L(Q) apple L(P) ifq P, I Superadditivity: L(P)+L(Q) apple L(P + Q), Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 6/15

12 Minkowski length: Properties Simple Properties: I Invariance: L(P) isagl(d, Z)-invariant, I Monotonicity: L(Q) apple L(P) ifq P, I Superadditivity: L(P)+L(Q) apple L(P + Q), I Bound: P \ Z d apple (L(P) + 1) d In particular, if P is strongly indecomposable then P \ Z d apple 2 d....think about P \ (Z/2Z) d... Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 6/15

13 Strongly indecomposable lattice polytopes in small dimension dim P = 1 primitive lattice segments dim P = 2 two classes of triangles Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 7/15

14 Strongly indecomposable lattice polytopes in small dimension dim P = 1 primitive lattice segments dim P = 2 two classes of triangles Theorem (Josh Whitney, 2010) Let P be strongly indecomposable, dim P =3. Then I P may have 4, 5, or 6 vertices only I There are infinite families of such P: I empty and clean tetrahedra I empty clean and non-clean double pyramids I empty clean and non-clean 6 vertex polytopes I There are = 107 classes of non-empty P Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 7/15

15 Back to counting rational points on hypersurfaces Theorem (S-Soprunova, 08) Let P be a lattice polygon with L = L(P). Then for q > (P). N q (P) apple L(q 1) + 2 p q 1 Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 8/15

16 Back to counting rational points on hypersurfaces Theorem (S-Soprunova, 08) Let P be a lattice polygon with L = L(P). Then for q > (P). N q (P) apple L(q 1) + 2 p q 1 Theorem (Whitney, 10) Let P be a lattice polytope in R 3 with L = L(P). Thenforq > (P) N q (P) apple N q (Q 1 )+ + N q (Q L ), for any Q Q L P and N q (Q i ) have explicit formulas based on the classification of 3-dim strongly indecomposable polytopes. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 8/15

17 Back to counting rational points on hypersurfaces Theorem (S-Soprunova, 08) Let P be a lattice polygon with L = L(P). Then for q > (P). N q (P) apple L(q 1) + 2 p q 1 Theorem (Whitney, 10) Let P be a lattice polytope in R 3 with L = L(P). Thenforq > (P) N q (P) apple N q (Q 1 )+ + N q (Q L ), for any Q Q L P and N q (Q i ) have explicit formulas based on the classification of 3-dim strongly indecomposable polytopes. Theorem (Beckwith Grimm Soprunova Weaver, 12) The total number of interior points in the Q i in any maximal decomposition is at most 4. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 8/15

18 How to compute L(P)? Let L = L(P). The maximal decompositions Q Q L P form a poset with respect to inclusion (up to a lattice translation). Minimal elements are smallest maximal decompositions. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 9/15

19 How to compute L(P)? Let L = L(P). The maximal decompositions Q Q L P form a poset with respect to inclusion (up to a lattice translation). Minimal elements are smallest maximal decompositions. Proposition Every smallest maximal decomposition is a lattice zonotope Z min P with at most 2 d 1 distinct direction vectors. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 9/15

20 How to compute L(P)? Let L = L(P). The maximal decompositions Q Q L P form a poset with respect to inclusion (up to a lattice translation). Minimal elements are smallest maximal decompositions. Proposition Every smallest maximal decomposition is a lattice zonotope Z min P with at most 2 d 1 distinct direction vectors. Reason: The direction vectors v 1,...,v k are non-zero mod 2. If k 2 d then v i + v j =2v for some i < j. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 9/15

21 Relation to Lattice Diameter Let P be a lattice polytope in R d. Definition: The largest number of lattice polytopes of positive dimension whose Minkowski sum is at most n-dimensional and is contained in P is called the n-th Minkowski length: L n (P) =max{l 2 N Q = Q Q L P, dim Q i > 0, dim Q apple n}. Clearly L 1 (P) apple L 2 (P) apple apple L d (P) =L(P). Note that L 1 (P) =lattice diameter of P. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 10/15

22 Relation to Lattice Diameter Let P be a lattice polytope in R d. Definition: The largest number of lattice polytopes of positive dimension whose Minkowski sum is at most n-dimensional and is contained in P is called the n-th Minkowski length: L n (P) =max{l 2 N Q = Q Q L P, dim Q i > 0, dim Q apple n}. Clearly L 1 (P) apple L 2 (P) apple apple L d (P) =L(P). Note that L 1 (P) =lattice diameter of P. Example I L 1 (P) = 2, L 2 (P) =3 I L n (t )=t for any n 2 N Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 10/15

23 Rational Minkowski length Main Question: How does L n (tp) behave as a function of t 2 N? Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 11/15

24 Rational Minkowski length Main Question: How does L n (tp) behave as a function of t 2 N? Definition: Define rational (asymptotic) Minkowski length: We put (P) = d (P). n(p) :=sup t2n L n (tp). t Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 11/15

25 Rational Minkowski length Main Question: How does L n (tp) behave as a function of t 2 N? Definition: Define rational (asymptotic) Minkowski length: We put (P) = d (P). In particular, n(p) :=sup t2n L n (tp). t 1(P) is the rational diameter, that is 1(P) =max{s P (v) primitive v 2 Z d }, where s P (v) is the diameter of P in the direction of v (relative to vz Z d ). This implies L 1 (tp) =b 1 (tp)c = b 1 (P)tc, which is quasi-linear (i.e. quasi-polynomial in t with linear constituencies). Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 11/15

26 Rational Minkowski length Theorem (S-Soprunova 16) Let P be a lattice polytope in R d.thereexistsk 2 N such that (P) = L(kP). k The smallest such k is the period of P. Corollary: (tp) =t (P) foranyt 2 N. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 12/15

27 Rational Minkowski length Theorem (S-Soprunova 16) Let P be a lattice polytope in R d.thereexistsk 2 N such that (P) = L(kP). k The smallest such k is the period of P. Corollary: (tp) =t (P) foranyt 2 N. Sketch of the proof: I (P) equals the supremum of the normalized perimeter p(z) of all rational zonotopes Z P. I The number of directions for the summands of Z min is bounded by 2 d 1 Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 12/15

28 Rational Minkowski length Theorem (S-Soprunova 16) Let P be a lattice polytope in R d.thereexistsk 2 N such that (P) = L(kP). k The smallest such k is the period of P. Corollary: (tp) =t (P) foranyt 2 N. Sketch of the proof: I There are only finitely many collections of directions for Z min for which p(z) is close to (P) I p(z) is the maximum of a linear function on a finite set of rational polytopes Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 12/15

29 Eventual Quasi-linearity of Minkowski length Theorem (S-Soprunova 16) Let P be a lattice polytope in R d with period k. ThenL(tP) is eventually quasi-linear in t, thatis,thereexistc r 2 Z for 0 apple r < k such that for all t 0 j t k L(tP) =k (P) + c r, whenever t r mod k. k Moreover, L(rP) apple c r apple r (P). Remark: The same statement holds for L n (tp) foranyn 2 N. Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 13/15

30 Example P 1-st Minkowski length (lattice diameter) I We have L 1 (P) =2and 1 (P) = 5 2 I L 1 (tp) =bt 1 (P)c =5b t 2c + {0, 2} 2P 3P Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 14/15

31 Example P 2P 1-st Minkowski length (lattice diameter) I We have L 1 (P) =2and 1 (P) = 5 2 I L 1 (tp) =bt 1 (P)c =5b t 2c + {0, 2} 2-nd Minkowski length I We have (P) apple 2Vol 2(P) w(p) for any polygon P I Here (P) = 10 3 and P has period k = 3. I L(tP) = 10b t 3 c+{0,3,6} 3P Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 14/15

32 Questions 1. As we saw earlier L n (t )=L 1 (t )foranyn 2 N. Itturns out that L n (T )=L 1 (T ) for any triangle in R 2.Doesthis hold for simplices in any dimension? Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 15/15

33 Questions 1. As we saw earlier L n (t )=L 1 (t )foranyn 2 N. Itturns out that L n (T )=L 1 (T ) for any triangle in R 2.Doesthis hold for simplices in any dimension? 2. Can the bound P \ Z d apple (L(P) + 1) d be improved? Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 15/15

34 Questions 1. As we saw earlier L n (t )=L 1 (t )foranyn 2 N. Itturns out that L n (T )=L 1 (T ) for any triangle in R 2.Doesthis hold for simplices in any dimension? 2. Can the bound P \ Z d apple (L(P) + 1) d be improved? 3. The above bound is sharp for strongly indecomposable polytopes P. What is the largest number of interior lattice points in strongly indecomposable P? Isit2 d (d + 1)? Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 15/15

35 Questions 1. As we saw earlier L n (t )=L 1 (t )foranyn 2 N. Itturns out that L n (T )=L 1 (T ) for any triangle in R 2.Doesthis hold for simplices in any dimension? 2. Can the bound P \ Z d apple (L(P) + 1) d be improved? 3. The above bound is sharp for strongly indecomposable polytopes P. What is the largest number of interior lattice points in strongly indecomposable P? Isit2 d (d + 1)? The End Ivan Soprunov (with Jenya Soprunova), Cleveland State University Minkowski Length of Lattice Polytopes 15/15