Tropical Volume by Tropical Ehrhart Polynomials Georg Loho London - PowerPoint PPT Presentation

Tropical Volume by Tropical Ehrhart Polynomials Georg Loho London School of Economics Matthias Schymura ´ Ecole Polytechnique F´ ed´ erale de Lausanne September 24, 2019 Discrete Geometry with a View on Symplectic and Tropical Geometry K¨ oln, Deutschland

Introduction Tropical semiring is T = ( R ∪ {−∞} , ⊕ , ⊙ ) with a ⊕ b = max { a , b } and a ⊙ b = a + b . −∞

Introduction Tropical semiring is T = ( R ∪ {−∞} , ⊕ , ⊙ ) with a ⊕ b = max { a , b } and a ⊙ b = a + b . The tropical convex hull of V = ( v 1 , . . . , v m ) ∈ T d × m is given by m m � � � � tconv( V ) = λ j ⊙ v j : λ 1 , . . . , λ m ∈ T , λ j = 0 . j =1 j =1 −∞

Introduction Tropical semiring is T = ( R ∪ {−∞} , ⊕ , ⊙ ) with a ⊕ b = max { a , b } and a ⊙ b = a + b . The tropical convex hull of V = ( v 1 , . . . , v m ) ∈ T d × m is given by m m � � � � tconv( V ) = λ j ⊙ v j : λ 1 , . . . , λ m ∈ T , λ j = 0 . j =1 j =1 P := tconv( V ) is called a tropical polytope . −∞

Introduction Tropical semiring is T = ( R ∪ {−∞} , ⊕ , ⊙ ) with a ⊕ b = max { a , b } and a ⊙ b = a + b . The tropical convex hull of V = ( v 1 , . . . , v m ) ∈ T d × m is given by m m � � � � tconv( V ) = λ j ⊙ v j : λ 1 , . . . , λ m ∈ T , λ j = 0 . j =1 j =1 P := tconv( V ) is called a tropical polytope . Recent studies show that metric tropical concepts are useful: log-barrier methods (Allamigeon, Benchimol, Gaubert & Joswig, 2018) tropical Voronoi diagrams (Criado, Joswig & Santos, 2019) tropical isodiametric inequality (Depersin, Gaubert & Joswig, 2017) −∞

Introduction Tropical semiring is T = ( R ∪ {−∞} , ⊕ , ⊙ ) with a ⊕ b = max { a , b } and a ⊙ b = a + b . The tropical convex hull of V = ( v 1 , . . . , v m ) ∈ T d × m is given by m m � � � � tconv( V ) = λ j ⊙ v j : λ 1 , . . . , λ m ∈ T , λ j = 0 . j =1 j =1 P := tconv( V ) is called a tropical polytope . Recent studies show that metric tropical concepts are useful: log-barrier methods (Allamigeon, Benchimol, Gaubert & Joswig, 2018) tropical Voronoi diagrams (Criado, Joswig & Santos, 2019) tropical isodiametric inequality (Depersin, Gaubert & Joswig, 2017) Main goal: Identify an instrinsic volume concept for tropical polytopes. −∞

Review of classical volume Let P ⊆ R d be a polytope. The classical volume concept for P is the Lebesgue measure: � vol( P ) := 1 dx . P π

Review of classical volume Let P ⊆ R d be a polytope. The classical volume concept for P is the Lebesgue measure: � vol( P ) := 1 dx . P First discretization: P ∩ 1 k Z d � kP ∩ Z d � � � # # vol( P ) = lim = lim k d k d k →∞ k →∞ π

Review of classical volume Let P ⊆ R d be a polytope. The classical volume concept for P is the Lebesgue measure: � vol( P ) := 1 dx . P First discretization: P ∩ 1 k Z d � kP ∩ Z d � � � # # vol( P ) = lim = lim k d k d k →∞ k →∞ Second discretization: Theorem (Ehrhart, 1967) If P is an integral polytope, that is, all vertices are from Z d , then = c d ( P ) k d + c d − 1 ( P ) k d − 1 + . . . + c 1 ( P ) k + c 0 ( P ) , � kP ∩ Z d � # for k ∈ N . In particular, c d ( P ) = vol( P ). π

Review of classical volume Let P ⊆ R d be a polytope. The classical volume concept for P is the Lebesgue measure: � vol( P ) := 1 dx . P First discretization: P ∩ 1 k Z d � kP ∩ Z d � � � # # vol( P ) = lim = lim k d k d k →∞ k →∞ Second discretization: Theorem (Ehrhart, 1967) If P is an integral polytope, that is, all vertices are from Z d , then = c d ( P ) k d + c d − 1 ( P ) k d − 1 + . . . + c 1 ( P ) k + c 0 ( P ) , � kP ∩ Z d � # for k ∈ N . In particular, c d ( P ) = vol( P ). Idea: Retrieve concept of tropical volume by turning this around – tropically. π

❘ Tropical lattices and tropical lattice polytopes Natural idea: Tropical integers could be TN := Z ≥ 0 ∪ {−∞} . 42

❘ Tropical lattices and tropical lattice polytopes Natural idea: Tropical integers could be TN := Z ≥ 0 ∪ {−∞} . Gaubert & MacCaig, 2017: Counting points from TN d in a tropical polytope is #P-hard. Computing vol( P ) is #P-hard. 42

❘ Tropical lattices and tropical lattice polytopes Natural idea: Tropical integers could be TN := Z ≥ 0 ∪ {−∞} . Gaubert & MacCaig, 2017: Counting points from TN d in a tropical polytope is #P-hard. Computing vol( P ) is #P-hard. Definition (Tropical lattice polytope) If all vertices of a tropical polytope P ⊆ T d are contained in TN d , then P is called a tropical lattice polytope . 42

❘ Tropical lattices and tropical lattice polytopes Natural idea: Tropical integers could be TN := Z ≥ 0 ∪ {−∞} . Gaubert & MacCaig, 2017: Counting points from TN d in a tropical polytope is #P-hard. Computing vol( P ) is #P-hard. Definition (Tropical lattice polytope) If all vertices of a tropical polytope P ⊆ T d are contained in TN d , then P is called a tropical lattice polytope . However, for an intrinsic volume definition and tropical Ehrhart theory, TN d is too rough. 42

❘ Tropical lattices and tropical lattice polytopes Natural idea: Tropical integers could be TN := Z ≥ 0 ∪ {−∞} . Gaubert & MacCaig, 2017: Counting points from TN d in a tropical polytope is #P-hard. Computing vol( P ) is #P-hard. Definition (Tropical lattice polytope) If all vertices of a tropical polytope P ⊆ T d are contained in TN d , then P is called a tropical lattice polytope . However, for an intrinsic volume definition and tropical Ehrhart theory, TN d is too rough. Transition from (+ , · )-convexity, over (max , · )-convexity, to (max , +)-convexity motivates: Definition (Tropical b -lattice) For b ∈ N ≥ 2 , the tropical b-lattice in T d is defined as log b ( Z ≥ 0 ) d := { (log b ( x 1 ) , . . . , log b ( x d )) : x 1 , . . . , x d ∈ Z ≥ 0 } . 42

Tropical lattices and tropical lattice polytopes Natural idea: Tropical integers could be TN := Z ≥ 0 ∪ {−∞} . Gaubert & MacCaig, 2017: Counting points from TN d in a tropical polytope is #P-hard. Computing vol( P ) is #P-hard. Definition (Tropical lattice polytope) If all vertices of a tropical polytope P ⊆ T d are contained in TN d , then P is called a tropical lattice polytope . However, for an intrinsic volume definition and tropical Ehrhart theory, TN d is too rough. Transition from (+ , · )-convexity, over (max , · )-convexity, to (max , +)-convexity motivates: Definition (Tropical b -lattice) For b ∈ N ≥ 2 , the tropical b-lattice in T d is defined as log b ( Z ≥ 0 ) d := { (log b ( x 1 ) , . . . , log b ( x d )) : x 1 , . . . , x d ∈ Z ≥ 0 } . ❘ TN d ⊆ � b ∈ N ≥ 2 log b ( Z ≥ 0 ) d 42

Tropical Ehrhart polynomial Theorem (L & Schymura, 2019+) Let b ∈ N ≥ 2 and let P ⊆ T d be a tropical lattice polytope. Then, for k ∈ Z ≥ 0 , the tropical lattice point enumerator L b � ( k ⊙ P ) ∩ log b ( Z ≥ 0 ) d � P ( k ) = # agrees with a polynomial in b k of degree at most d. 4/12

Tropical Ehrhart polynomial Theorem (L & Schymura, 2019+) Let b ∈ N ≥ 2 and let P ⊆ T d be a tropical lattice polytope. Then, for k ∈ Z ≥ 0 , the tropical lattice point enumerator L b � ( k ⊙ P ) ∩ log b ( Z ≥ 0 ) d � P ( k ) = # agrees with a polynomial in b k of degree at most d. We write d ( P )( b k ) d + c b d − 1 ( P )( b k ) d − 1 + . . . + c b 1 ( P ) b k + c b L b P ( k ) = c b 0 ( P ) , i ( P ) the i th tropical Ehrhart coefficient of P . and call c b 4/12

Tropical Ehrhart polynomial Theorem (L & Schymura, 2019+) Let b ∈ N ≥ 2 and let P ⊆ T d be a tropical lattice polytope. Then, for k ∈ Z ≥ 0 , the tropical lattice point enumerator L b ( k ⊙ P ) ∩ log b ( Z ≥ 0 ) d � � P ( k ) = # agrees with a polynomial in b k of degree at most d. We write d ( P )( b k ) d + c b d − 1 ( P )( b k ) d − 1 + . . . + c b 1 ( P ) b k + c b L b P ( k ) = c b 0 ( P ) , i ( P ) the i th tropical Ehrhart coefficient of P . and call c b 3 log 2 (7) �� �� 0 0 1 b = 2, P = tconv log 2 (6) 0 1 1 log 2 (5) 2 log 2 (3) 1 P 0 1 log 2 (3) 2 log 2 (5) 4/12

Tropical Ehrhart polynomial Theorem (L & Schymura, 2019+) Let b ∈ N ≥ 2 and let P ⊆ T d be a tropical lattice polytope. Then, for k ∈ Z ≥ 0 , the tropical lattice point enumerator L b ( k ⊙ P ) ∩ log b ( Z ≥ 0 ) d � � P ( k ) = # agrees with a polynomial in b k of degree at most d. We write d ( P )( b k ) d + c b d − 1 ( P )( b k ) d − 1 + . . . + c b 1 ( P ) b k + c b L b P ( k ) = c b 0 ( P ) , i ( P ) the i th tropical Ehrhart coefficient of P . and call c b 3 log 2 (7) �� �� 0 0 1 b = 2, P = tconv log 2 (6) 0 1 1 log 2 (5) 2 log 2 (3) L 2 P (0) = 3 1 P = 0 ⊙ P 0 1 log 2 (3) 2 log 2 (5) 4/12

Tropical Ehrhart polynomial Theorem (L & Schymura, 2019+) Let b ∈ N ≥ 2 and let P ⊆ T d be a tropical lattice polytope. Then, for k ∈ Z ≥ 0 , the tropical lattice point enumerator L b ( k ⊙ P ) ∩ log b ( Z ≥ 0 ) d � � P ( k ) = # agrees with a polynomial in b k of degree at most d. We write d ( P )( b k ) d + c b d − 1 ( P )( b k ) d − 1 + . . . + c b 1 ( P ) b k + c b L b P ( k ) = c b 0 ( P ) , i ( P ) the i th tropical Ehrhart coefficient of P . and call c b 3 log 2 (7) �� �� 0 0 1 b = 2, P = tconv log 2 (6) 0 1 1 log 2 (5) 2 log 2 (3) L 2 P (0) = 3, L 2 P (1) = 6 1 ⊙ P 1 0 1 log 2 (3) 2 log 2 (5) 4/12