DedekindCarlitz Polynomials as Lattice-Point Enumerators in - PowerPoint PPT Presentation

Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck (San Francisco State University) Christian Haase (Freie Universit¨ at Berlin) Asia Matthews (Queen’s University) arXiv:0710.1323 math.sfsu.edu/beck

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Dedekind Sums Let � { x } − ⌊ x ⌋ − 1 if x / ∈ Z , 2 ( ( x ) ) := 0 if x ∈ Z , and define for positive integers a and b the Dedekind sum b − 1 � � ka � � � � k � � � s ( a, b ) := b b k =0 Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 3

Dedekind Sums Let � { x } − ⌊ x ⌋ − 1 if x / ∈ Z , 2 ( ( x ) ) := 0 if x ∈ Z , and define for positive integers a and b the Dedekind sum b − 1 � � ka � � � � k � � � s ( a, b ) := b b k =0 b − 1 − 1 � ka � � = ( k − 1) + easy( a, b ) . b b k =1 Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 3

Dedekind Sums Let � { x } − ⌊ x ⌋ − 1 if x / ∈ Z , 2 ( ( x ) ) := 0 if x ∈ Z , and define for positive integers a and b the Dedekind sum b − 1 � � ka � � � � k � � � s ( a, b ) := b b k =0 b − 1 − 1 � ka � � = ( k − 1) + easy( a, b ) . b b k =1 Since their introduction in the 1880’s, the Dedekind sum and its generalizations have intrigued mathematicians from various areas such as analytic and algebraic number theory, topology, algebraic and combinatorial geometry, and algorithmic complexity. Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 3

Dedekind–Carlitz Polynomials In the 1970’s, Leonard Carlitz introduced the following polynomial generalization of the Dedekind sum: b − 1 u ⌊ ka b ⌋ v k − 1 . � c ( u, v ; a, b ) := k =1 Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 4

Dedekind–Carlitz Polynomials In the 1970’s, Leonard Carlitz introduced the following polynomial generalization of the Dedekind sum: b − 1 u ⌊ ka b ⌋ v k − 1 . � c ( u, v ; a, b ) := k =1 Carlitz proved the following reciprocity law if a and b are relatively prime: ( v − 1) c ( u, v ; a, b ) + ( u − 1) c ( v, u ; b, a ) = u a − 1 v b − 1 − 1 . Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 4

Dedekind–Carlitz Polynomials In the 1970’s, Leonard Carlitz introduced the following polynomial generalization of the Dedekind sum: b − 1 u ⌊ ka b ⌋ v k − 1 . � c ( u, v ; a, b ) := k =1 Carlitz proved the following reciprocity law if a and b are relatively prime: ( v − 1) c ( u, v ; a, b ) + ( u − 1) c ( v, u ; b, a ) = u a − 1 v b − 1 − 1 . Applying u ∂u twice and v ∂v once gives Dedekind’s reciprocity law � a � s ( a, b ) + s ( b, a ) = − 1 4 + 1 b + 1 ab + b . 12 a Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 4

Enter Polyhedral Geometry Decompose the first quadrant R 2 ≥ 0 into the two cones K 1 = { λ 1 (0 , 1) + λ 2 ( a, b ) : λ 1 , λ 2 ≥ 0 } , K 2 = { λ 1 (1 , 0) + λ 2 ( a, b ) : λ 1 ≥ 0 , λ 2 > 0 } . Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 5

Enter Polyhedral Geometry Decompose the first quadrant R 2 ≥ 0 into the two cones K 1 = { λ 1 (0 , 1) + λ 2 ( a, b ) : λ 1 , λ 2 ≥ 0 } , K 2 = { λ 1 (1 , 0) + λ 2 ( a, b ) : λ 1 ≥ 0 , λ 2 > 0 } . Let’s compute the integer-point transforms � u m v n σ K 1 ( u, v ) := ( m,n ) ∈K 1 ∩ Z 2 Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 5

Enter Polyhedral Geometry Decompose the first quadrant R 2 ≥ 0 into the two cones K 1 = { λ 1 (0 , 1) + λ 2 ( a, b ) : λ 1 , λ 2 ≥ 0 } , K 2 = { λ 1 (1 , 0) + λ 2 ( a, b ) : λ 1 ≥ 0 , λ 2 > 0 } . Let’s compute the integer-point transforms     u m v n = σ Π 1 ( u, v ) � � � v j u ka v kb σ K 1 ( u, v ) :=   ( m,n ) ∈K 1 ∩ Z 2 j ≥ 0 k ≥ 0 σ Π 1 ( u, v ) = (1 − v ) (1 − u a v b ) , where Π 1 is the fundamental parallelogram of K 1 : Π 1 = { λ 1 (0 , 1) + λ 2 ( a, b ) : 0 ≤ λ 1 , λ 2 < 1 } . Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 5

Carlitz Reciprocity The integer points in this parallelogram are � � � kb � � � Π 1 ∩ Z 2 = (0 , 0) , k, + 1 : 1 ≤ k ≤ a − 1 , k ∈ Z . a Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 6

Carlitz Reciprocity The integer points in this parallelogram are � � � kb � � � Π 1 ∩ Z 2 = (0 , 0) , k, + 1 : 1 ≤ k ≤ a − 1 , k ∈ Z , a from which we obtain k =1 u k v ⌊ kb a ⌋ +1 σ K 1 ( u, v ) = 1 + � a − 1 = 1 + uv c ( v, u ; b, a ) ( v − 1) ( u a v b − 1) . (1 − v )(1 − u a v b ) Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 6

Carlitz Reciprocity The integer points in this parallelogram are � � � kb � � � Π 1 ∩ Z 2 = (0 , 0) , k, + 1 : 1 ≤ k ≤ a − 1 , k ∈ Z , a from which we obtain k =1 u k v ⌊ kb a ⌋ +1 σ K 1 ( u, v ) = 1 + � a − 1 = 1 + uv c ( v, u ; b, a ) ( v − 1) ( u a v b − 1) . (1 − v )(1 − u a v b ) u + uv c( u,v ; a,b ) Analogously, one computes σ K 2 ( u, v ) = and Carlitz’s ( u − 1) ( u a v b − 1 ) reciprocity law follows from 1 σ K 1 ( u, v ) + σ K 2 ( u, v ) = σ R 2 ≥ 0 ( u, v ) = (1 − u )(1 − v ) . Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 6

Higher Dimensions Our proof has a natural generalization to the higher-dimensional Dedekind– Carlitz polynomials — � kan − 1 a n − 1 j ka 1 k j ka 2 k an � an an u k − 1 c ( u 1 , u 2 , . . . , u n ; a 1 , a 2 , . . . , a n ) := u u · · · u , 1 2 n − 1 n k =1 where u 1 , u 2 , . . . , u n are indeterminates and a 1 , a 2 , . . . , a n are positive integers. Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 7

Higher Dimensions Our proof has a natural generalization to the higher-dimensional Dedekind– Carlitz polynomials — � kan − 1 a n − 1 j ka 1 k j ka 2 k an � an an u k − 1 c ( u 1 , u 2 , . . . , u n ; a 1 , a 2 , . . . , a n ) := u u · · · u , 1 2 n − 1 n k =1 where u 1 , u 2 , . . . , u n are indeterminates and a 1 , a 2 , . . . , a n are positive integers. Berndt–Dieter proved that if a 1 , a 2 , . . . , a n are pairwise relatively prime then ( u n − 1) c ( u 1 , u 2 , . . . , u n ; a 1 , a 2 , . . . , a n ) + ( u n − 1 − 1) c ( u n , u 1 , . . . , u n − 2 , u n − 1 ; a n , a 1 , . . . , a n − 2 , a n − 1 ) + · · · + ( u 1 − 1) c ( u 2 , u 3 , . . . , u n , u 1 ; a 2 , a 3 , . . . , a n , a 1 ) = u a 1 − 1 u a 2 − 1 · · · u a n − 1 − 1 . 1 2 n Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 7

Higher Dimensions Our proof has a natural generalization to the higher-dimensional Dedekind– Carlitz polynomials — � kan − 1 a n − 1 j ka 1 k j ka 2 k an � an an u k − 1 c ( u 1 , u 2 , . . . , u n ; a 1 , a 2 , . . . , a n ) := u u · · · u , 1 2 n − 1 n k =1 where u 1 , u 2 , . . . , u n are indeterminates and a 1 , a 2 , . . . , a n are positive integers. Berndt–Dieter proved that if a 1 , a 2 , . . . , a n are pairwise relatively prime then ( u n − 1) c ( u 1 , u 2 , . . . , u n ; a 1 , a 2 , . . . , a n ) + ( u n − 1 − 1) c ( u n , u 1 , . . . , u n − 2 , u n − 1 ; a n , a 1 , . . . , a n − 2 , a n − 1 ) + · · · + ( u 1 − 1) c ( u 2 , u 3 , . . . , u n , u 1 ; a 2 , a 3 , . . . , a n , a 1 ) = u a 1 − 1 u a 2 − 1 · · · u a n − 1 − 1 . 1 2 n We could shift the cones involved in our proofs by a fixed vector. This gives rise to shifts in the greatest-integer functions, and the resulting Carlitz sums are polynomial analogues of Dedekind–Rademacher sums. Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 7

Computational Complexity Dedekind reciprocity immediately yields an efficient algorithm to compute Dedekind sums; however, we do not know how to derive a similar complexity statement from Carlitz reciprocity. Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 8

Computational Complexity Dedekind reciprocity immediately yields an efficient algorithm to compute Dedekind sums; however, we do not know how to derive a similar complexity statement from Carlitz reciprocity. Fortunately, Barvinok proved in the 1990’s that in fixed dimension, the integer-point transform σ P ( z 1 , z 2 , . . . , z d ) of a rational polyhedron P can be computed as a sum of rational functions in z 1 , z 2 , . . . , z d in time polynomial in the input size of P . Dedekind–Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra Matthias Beck 8