Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices - PowerPoint PPT Presentation

Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck & Serkan Ho¸ sten San Francisco State University math.sfsu.edu/beck Math Research Letters

Growth Series of Lattices L ⊂ R d – lattice of rank r M – subset that generates L as a monoid Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 2

Growth Series of Lattices L ⊂ R d – lattice of rank r M – subset that generates L as a monoid S ( n ) – number of elements in L with word length n (with respect to M ) n ≥ 0 S ( n ) x n – growth series of ( L , M ) G ( x ) := � h ( x ) G ( x ) = (1 − x ) r where h is the coordinator polynomial of ( L , M ) Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 2

Growth Series of Lattices L ⊂ R d – lattice of rank r M – subset that generates L as a monoid S ( n ) – number of elements in L with word length n (with respect to M ) n ≥ 0 S ( n ) x n – growth series of ( L , M ) G ( x ) := � h ( x ) G ( x ) = (1 − x ) r where h is the coordinator polynomial of ( L , M ) • • • • • • • • • • ✡❏ ❏ ✡ • • • ◗ • • ◗ ✑ ✑ � � � � ☎☎ ✥ ✥ • • � • • • • • • • • Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 2

Growth Series of Lattices L ⊂ R d – lattice of rank r M – subset that generates L as a monoid S ( n ) – number of elements in L with word length n (with respect to M ) n ≥ 0 S ( n ) x n – growth series of ( L , M ) G ( x ) := � h ( x ) G ( x ) = (1 − x ) r where h is the coordinator polynomial of ( L , M ) • • • • • • • • • • • • • • • • • • • • ✡❏ ❏ ✡ ✡❏ ❏ ✡ ✑ • • • ◗ • • • • ✑ • ◗ ◗ • • ◗ ✑ ✑ ✑ ◗ ◗ ✑ � � � � ☎☎ ❏✡ ✡ ❏ ✥ ✥ • • • • • • • � • • • • • • • • • • • • • Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 2

Cyclotomic Lattices L = Z [ e 2 πi/m ] ∼ = Z ϕ ( m ) M – all m th roots of unity (suitably identified in R ϕ ( m ) ) h m – coordinator polynomial of Z [ e 2 πi/m ] Initiated by Parker, motivated by error-correcting codes and random walks Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 3

Cyclotomic Lattices L = Z [ e 2 πi/m ] ∼ = Z ϕ ( m ) M – all m th roots of unity (suitably identified in R ϕ ( m ) ) h m – coordinator polynomial of Z [ e 2 πi/m ] Initiated by Parker, motivated by error-correcting codes and random walks Theorem (Kløve–Parker) The coordinator polynomial of Z [ e 2 πi/p ] , where p is prime, equals h p ( x ) = x p − 1 + x p − 2 + · · · + 1 . Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 3

Cyclotomic Lattices L = Z [ e 2 πi/m ] ∼ = Z ϕ ( m ) M – all m th roots of unity (suitably identified in R ϕ ( m ) ) h m – coordinator polynomial of Z [ e 2 πi/m ] Initiated by Parker, motivated by error-correcting codes and random walks Theorem (Kløve–Parker) The coordinator polynomial of Z [ e 2 πi/p ] , where p is prime, equals h p ( x ) = x p − 1 + x p − 2 + · · · + 1 . Conjectures (Parker) √ m for a palindromic polynomial g of degree ϕ ( √ m ) . m (1) h m ( x ) = g ( x ) (2) h 2 p ( x ) = � p − 3 � x k + x p − 1 − k � � k � p � p − 1 + 2 p − 1 x 2 . 2 k =0 j =0 j � 1 + x 8 � � x + x 7 � � x 2 + x 6 � � x 3 + x 5 � +130 x 4 . (3) h 15 ( x ) = +7 +28 +79 Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 3

Main Results Conjectures (Parker) √ m for a palindromic polynomial g of degree ϕ ( √ m ) . m (1) h m ( x ) = g ( x ) (2) h 2 p ( x ) = � p − 3 � x k + x p − 1 − k � � k � p � p − 1 + 2 p − 1 x 2 2 k =0 j =0 j � 1 + x 8 � � x + x 7 � � x 2 + x 6 � � x 3 + x 5 � + 130 x 4 (3) h 15 ( x ) = + 7 + 28 + 79 Theorem (M B–Ho¸ sten) Suppose m is divisible by at most two odd primes. m √ m (1) h m ( x ) = h √ m ( x ) Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 4

Main Results Conjectures (Parker) √ m for a palindromic polynomial g of degree ϕ ( √ m ) . m (1) h m ( x ) = g ( x ) (2) h 2 p ( x ) = � p − 3 � x k + x p − 1 − k � � k � p � p − 1 + 2 p − 1 x 2 2 k =0 j =0 j � 1 + x 8 � � x + x 7 � � x 2 + x 6 � � x 3 + x 5 � + 130 x 4 (3) h 15 ( x ) = + 7 + 28 + 79 Theorem (M B–Ho¸ sten) Suppose m is divisible by at most two odd primes. m √ m (1) h m ( x ) = h √ m ( x ) (2) h √ m ( x ) is the h-polynomial of a simplicial polytope. Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 4

Main Results Conjectures (Parker) √ m for a palindromic polynomial g of degree ϕ ( √ m ) . m (1) h m ( x ) = g ( x ) (2) h 2 p ( x ) = � p − 3 � x k + x p − 1 − k � � k � p � p − 1 + 2 p − 1 x 2 2 k =0 j =0 j � 1 + x 8 � � x + x 7 � � x 2 + x 6 � � x 3 + x 5 � + 130 x 4 (3) h 15 ( x ) = + 7 + 28 + 79 Theorem (M B–Ho¸ sten) Suppose m is divisible by at most two odd primes. m √ m (1) h m ( x ) = h √ m ( x ) (2) h √ m ( x ) is the h-polynomial of a simplicial polytope. Corollary If m is divisible by at most two odd primes, then h √ m ( x ) is palindromic, unimodal, and has nonnegative integer coefficients. Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 4

Main Results Conjectures (Parker) √ m for a palindromic polynomial g of degree ϕ ( √ m ) . m (1) h m ( x ) = g ( x ) (2) h 2 p ( x ) = � p − 3 � x k + x p − 1 − k � � k � p � p − 1 + 2 p − 1 x 2 2 k =0 j =0 j � 1 + x 8 � � x + x 7 � � x 2 + x 6 � � x 3 + x 5 � + 130 x 4 (3) h 15 ( x ) = + 7 + 28 + 79 Theorem (M B–Ho¸ sten) Suppose m is divisible by at most two odd primes. m √ m (1) h m ( x ) = h √ m ( x ) (2) h √ m ( x ) is the h-polynomial of a simplicial polytope. Corollary If m is divisible by at most two odd primes, then h √ m ( x ) is palindromic, unimodal, and has nonnegative integer coefficients. Theorem (M B–Ho¸ sten) Parker’s Conjectures (2) & (3) are true. Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 4

Cyclotomic Polytopes We choose a specific basis for Z [ e 2 πi/m ] consisting of certain powers of e 2 πi/m which we then identify with the unit vectors in R ϕ ( m ) . The other powers of e 2 πi/m are integer linear combinations of this basis; hence they are lattice vectors in Z [ e 2 πi/m ] ⊂ R ϕ ( m ) . The m th cyclotomic polytope C m is the convex hull of all of these m lattice points in R ϕ ( m ) , which correspond to the m th roots of unity. Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 5

Cyclotomic Polytopes We choose a specific basis for Z [ e 2 πi/m ] consisting of certain powers of e 2 πi/m which we then identify with the unit vectors in R ϕ ( m ) . The other powers of e 2 πi/m are integer linear combinations of this basis; hence they are lattice vectors in Z [ e 2 πi/m ] ⊂ R ϕ ( m ) . The m th cyclotomic polytope C m is the convex hull of all of these m lattice points in R ϕ ( m ) , which correspond to the m th roots of unity. We will do this recursively in three steps: (1) m is prime (2) m is a prime power (3) m is the product of two coprime integers. Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 5

Cyclotomic Polytopes We choose a specific basis for Z [ e 2 πi/m ] consisting of certain powers of e 2 πi/m which we then identify with the unit vectors in R ϕ ( m ) . The other powers of e 2 πi/m are integer linear combinations of this basis; hence they are lattice vectors in Z [ e 2 πi/m ] ⊂ R ϕ ( m ) . The m th cyclotomic polytope C m is the convex hull of all of these m lattice points in R ϕ ( m ) , which correspond to the m th roots of unity. We will do this recursively in three steps: (1) m is prime (2) m is a prime power (3) m is the product of two coprime integers. When m = p is a prime number, let ζ = e 2 πi/p and fix the Z -basis 1 , ζ, ζ 2 , . . . , ζ p − 2 of the lattice Z [ ζ ] . Together with ζ p − 1 = − � p − 2 j =0 ζ j , these p elements form a monoid basis for Z [ ζ ] . We identify them with j =0 e j in R p − 1 and define the cyclotomic polytope e 0 , e 1 , . . . , e p − 2 , − � p − 2 C p ⊂ R p − 1 as the simplex � � p − 2 � C p = conv e 0 , e 1 , . . . , e p − 2 , − e i . i =0 Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 5

Cyclotomic Polytopes � � p − 2 � C p = conv e 0 , e 1 , . . . , e p − 2 , − e i . i =0 • • • • • • • • • • ✁❅ ✁ ❅ ✁ ❅ C 3 ✁ ❅ • • • • • ✁ ❅ ✟✟✟✟✟✟✟✟✟✟ ✁ ✁ ✁ ✁ • • ✁ • • • • • • • • Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices Matthias Beck 6