Enumerating integer points in polytopes: applications to number theory. Matthias Beck San Francisco State University math.sfsu.

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1 Enumerating integer points in polytopes: applications to number theory Matthias Beck San Francisco State University math.sfsu.edu/beck

2 It takes a village to count integer points. Alexander Barvinok

3 Outline Ehrhart theory Dedekind sums Coin-exchange problem of Frobenius Roots of Ehrhart polynomials All kinds of magic Enumerating integer-points in polytopes: applications to number theory Matthias Beck 3

4 Joint work with... Sinai Robins (Ehrhart formulas, Dedekind sums) Ricardo Diaz and Sinai Robins (Frobenius problem) Jesus De Loera, Mike Develin, Julian Pfeifle, and Richard Stanley (roots) Dennis Pixton (Birkhoff polytope) Moshe Cohen, Jessica Cuomo, Paul Gribelyuk (weak magic) Thomas Zaslavsky (strong magic) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 4

5 (Weak) semimagic squares H n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t Enumerating integer-points in polytopes: applications to number theory Matthias Beck 5

6 (Weak) semimagic squares H n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t Theorem (Ehrhart, Stanley 1973, conjectured by Anand-Dumir-Gupta 1966) H n (t) is a polynomial in t of degree (n 1) 2. This polynomial satisfies H n ( t) = ( 1) n 1 H n (t n) and H n ( 1) = = H n ( n + 1) = 0. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 5

7 (Weak) semimagic squares H n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t Theorem (Ehrhart, Stanley 1973, conjectured by Anand-Dumir-Gupta 1966) H n (t) is a polynomial in t of degree (n 1) 2. This polynomial satisfies H n ( t) = ( 1) n 1 H n (t n) and H n ( 1) = = H n ( n + 1) = 0. For example... H 1 (t) = 1 H 2 (t) = t + 1 (MacMahon 1905) H 3 (t) = 3 ( t+3 4 ) ( + t+2 ) 2 = 1 8 t t t t + 1 Enumerating integer-points in polytopes: applications to number theory Matthias Beck 5

8 Ehrhart theory Integral (convex) polytope P convex hull of finitely many points in Z d For t Z >0, let L P (t) := # ( tp Z d) = # ( P 1 t Zd) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 6

9 Ehrhart theory Integral (convex) polytope P convex hull of finitely many points in Z d For t Z >0, let L P (t) := # ( tp Z d) = # ( P 1 t Zd) Theorem (Ehrhart 1962) If P is an integral polytope, then... L P (t) and L P (t) are polynomials in t of degree dim P Leading term: vol(p ) (suitably normalized) (Macdonald 1970) L P ( t) = ( 1) dim P L P (t) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 6

10 Ehrhart theory Integral (convex) polytope P convex hull of finitely many points in Z d For t Z >0, let L P (t) := # ( tp Z d) = # ( P 1 t Zd) Theorem (Ehrhart 1962) If P is an integral polytope, then... L P (t) and L P (t) are polynomials in t of degree dim P Leading term: vol(p ) (suitably normalized) (Macdonald 1970) L P ( t) = ( 1) dim P L P (t) Alternative description of a polytope: P = { x R d : A x b } { x R d 0 : A x = b } Enumerating integer-points in polytopes: applications to number theory Matthias Beck 6

11 A magic example: the Birkhoff polytope B n = x 11 x 1n.. x n1... x nn R n2 0 : j x jk = 1 for all 1 k n k x jk = 1 for all 1 j n H n (t) = L Bn (t) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 7

12 A magic example: the Birkhoff polytope B n = x 11 x 1n.. x n1... x nn R n2 0 : j x jk = 1 for all 1 k n k x jk = 1 for all 1 j n H n (t) = L Bn (t) B n is a convex polytope of dimension (n 1) 2 Enumerating integer-points in polytopes: applications to number theory Matthias Beck 7

13 A magic example: the Birkhoff polytope B n = x 11 x 1n.. x n1... x nn R n2 0 : j x jk = 1 for all 1 k n k x jk = 1 for all 1 j n H n (t) = L Bn (t) B n is a convex polytope of dimension (n 1) 2 Vertices are the n n-permutation matrices. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 7

14 A magic example: the Birkhoff polytope B n = x 11 x 1n.. x n1... x nn R n2 0 : j x jk = 1 for all 1 k n k x jk = 1 for all 1 j n H n (t) = L Bn (t) B n is a convex polytope of dimension (n 1) 2 Vertices are the n n-permutation matrices. H n ( t) = ( 1) n 1 H n (t n) and H n ( 1) = = H n ( n + 1) = 0 follow with L B n (t) = H n (t n) and Ehrhart-Macdonald Reciprocity. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 7

15 A magic example: the Birkhoff polytope B n = x 11 x 1n.. x n1... x nn R n2 0 : j x jk = 1 for all 1 k n k x jk = 1 for all 1 j n H n (t) = L Bn (t) B n is a convex polytope of dimension (n 1) 2 Vertices are the n n-permutation matrices. H n ( t) = ( 1) n 1 H n (t n) and H n ( 1) = = H n ( n + 1) = 0 follow with L B n (t) = H n (t n) and Ehrhart-Macdonald Reciprocity. A close relative to B n appeared recently in connections with pseudomoments of the ζ-function (Conrey Gamburd 2005) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 7

16 Ehrhart theory Integral (convex) polytope P convex hull of finitely many points in Z d For t Z >0, let L P (t) := # ( tp Z d) = # ( P 1 t Zd) Theorem (Ehrhart 1962) If P is an integral polytope, then... L P (t) and L P (t) are polynomials in t of degree dim P Leading term: vol(p ) (suitably normalized) (Macdonald 1970) L P ( t) = ( 1) dim P L P (t) Alternative description of a polytope: P = { x R d : A x b } { x R d 0 : A x = b } Enumerating integer-points in polytopes: applications to number theory Matthias Beck 8

17 Ehrhart theory Rational (convex) polytope P convex hull of finitely many points in Q d For t Z >0, let L P (t) := # ( tp Z d) = # ( P 1 t Zd) Theorem (Ehrhart 1962) If P is an rational polytope, then... L P (t) and L P (t) are quasi-polynomials in t of degree dim P Leading term: vol(p ) (suitably normalized) (Macdonald 1970) L P ( t) = ( 1) dim P L P (t) Alternative description of a polytope: P = { x R d : A x b } { x R d 0 : A x = b } Quasi-polynomial c d (t) t d + c d 1 (t) t d c 0 (t) where c k (t) are periodic Enumerating integer-points in polytopes: applications to number theory Matthias Beck 8

18 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} (a = 7, b = 4, t = 23) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 9

19 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} (a = 7, b = 4, t = 23) L (t) = # { (m, n) Z 2 0 : am + bn t } Enumerating integer-points in polytopes: applications to number theory Matthias Beck 9

20 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} (a = 7, b = 4, t = 23) L (t) = # { (m, n) Z 2 0 : am + bn t } = # { (m, n, s) Z 3 0 : am + bn + s = t } Enumerating integer-points in polytopes: applications to number theory Matthias Beck 9

21 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} (a = 7, b = 4, t = 23) L (t) = # { (m, n) Z 2 0 : am + bn t } = # { (m, n, s) Z 3 0 : am + bn + s = t } = const 1 (1 x a ) (1 x b ) (1 x) x t Enumerating integer-points in polytopes: applications to number theory Matthias Beck 9

22 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} (a = 7, b = 4, t = 23) L (t) = # { (m, n) Z 2 0 : am + bn t } = # { (m, n, s) Z 3 0 : am + bn + s = t } = const = 1 2πi 1 (1 x a ) (1 x b ) (1 x) x t x =ɛ dx (1 x a ) (1 x b ) (1 x) x t+1 Enumerating integer-points in polytopes: applications to number theory Matthias Beck 9

23 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} 1 f(x) := (1 x a ) (1 x b ) (1 x) x t+1 L (t) = 1 2πi x =ɛ f dx Enumerating integer-points in polytopes: applications to number theory Matthias Beck 10

24 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} gcd (a, b) = 1 f(x) := 1 (1 x a ) (1 x b ) (1 x) x t+1 ξ a := e 2πi/a L (t) = 1 2πi x =ɛ f dx = Res x=1 (f) + a 1 k=1 Res x=ξ k a (f) + b 1 j=1 Res j x=ξ (f) b Enumerating integer-points in polytopes: applications to number theory Matthias Beck 10

25 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} gcd (a, b) = 1 f(x) := L (t) = 1 (1 x a ) (1 x b ) (1 x) x t+1 ξ a := e 2πi/a 1 2πi x =ɛ f dx = Res x=1 (f) + = t 2 2ab + t a a 1 k=1 a 1 k=1 ( 1 ab + 1 a + 1 ) b Res x=ξ k a (f) (1 ξa kb ) (1 ξa) k ξa kt + 1 b b 1 j=1 Res j x=ξ (f) b ( 3 a + 3 b a b + b a + 1 ) ab b 1 j=1 ( 1 ξ ja b 1 ) ( 1 ξ j b ) ξ jt b Enumerating integer-points in polytopes: applications to number theory Matthias Beck 10

26 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} (Pick s or) Ehrhart s Theorem implies that L has constant term L (0) = 1 1 a a 1 k=1 1 (1 ξa kb ) (1 ξa) k + 1 b = b 1 j=1 ( 1 ξ ja b ( 3 a + 3 b a b + b a + 1 ab 1 ) ( ) 1 ξ j b ) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 11

27 An example in dimension 2 := { (x, y) R 2 0 : ax + by 1} (Pick s or) Ehrhart s Theorem implies that L has constant term L (0) = 1 However... 1 a a 1 k=1 1 (1 ξa kb ) (1 ξa) k + 1 b = b 1 j=1 ( 1 ξ ja b ( 3 a + 3 b a b + b a + 1 ab 1 ) ( ) 1 ξ j b ) 1 a a 1 k=1 1 (1 ξa kb ) (1 ξa) k = 1 4a a 1 k=1 ( ) ( ) πkb πk cot cot a a + a 1 4a Enumerating integer-points in polytopes: applications to number theory Matthias Beck 11

28 Dedekind sums s (a, b) := 1 4b b 1 j=1 ( ) ( ) πja πj cot cot b b Since their introduction by Dedekind in the 1880 s, these sums and their generalizations have appeared in various areas such as analytic (transformation law of η-function) and algebraic number theory (class numbers), topology (group action on manifolds), combinatorial geometry (lattice point problems), and algorithmic complexity (random number generators). Enumerating integer-points in polytopes: applications to number theory Matthias Beck 12

29 Dedekind sums s (a, b) := 1 4b b 1 j=1 ( ) ( ) πja πj cot cot b b Since their introduction by Dedekind in the 1880 s, these sums and their generalizations have appeared in various areas such as analytic (transformation law of η-function) and algebraic number theory (class numbers), topology (group action on manifolds), combinatorial geometry (lattice point problems), and algorithmic complexity (random number generators). The identity L (0) = 1 implies... s (a, b) + s (b, a) = the Reciprocity Law for Dedekind sums. ( a b + 1 ab + b ) a Enumerating integer-points in polytopes: applications to number theory Matthias Beck 12

30 A 2-dimensional example in dimension 3 z t c := { (x, y, z) R 3 0 : ax + by + cz = 1} 1 c 1 b t b y 1 a t a x Enumerating integer-points in polytopes: applications to number theory Matthias Beck 13

31 t c z A 2-dimensional example in dimension 3 := { (x, y, z) R 3 0 : ax + by + cz = 1} 1 c gcd (a, b) = gcd (b, c) = gcd (c, a) = 1 1 b t b t a x 1 a = t2 2abc + t a a 1 k=1 L (t) = 1 2πi ( 1 ab + 1 ac + 1 ) bc y x =ɛ (1 ξa kb ) (1 ξa kc ) ξa kt + 1 b dx (1 x a ) (1 x b ) (1 x c ) x t+1 ( 3 a + 3 b + 3 c + a bc + b ac + c ) ab + 1 c b 1 k=1 c 1 k=1 ( 1 ξ kc b 1 ) ( 1 ξ ka b ) ξ kt b 1 (1 ξc ka ) (1 ξc kb ) ξc kt Enumerating integer-points in polytopes: applications to number theory Matthias Beck 13

32 More Dedekind sums s (a, b; c) := 1 4c c 1 j=1 ( ) ( ) πja πjb cot cot c c The identity L (0) = 1 implies Rademacher s Reciprocity Law s (a, b; c) + s (b, c; a) + s (c, a; b) = ( a bc + b ca + c ) ab Enumerating integer-points in polytopes: applications to number theory Matthias Beck 14

33 More Dedekind sums s (a, b; c) := 1 4c c 1 j=1 ( ) ( ) πja πjb cot cot c c The identity L (0) = 1 implies Rademacher s Reciprocity Law s (a, b; c) + s (b, c; a) + s (c, a; b) = ( a bc + b ca + c ) ab Moreover, t = { (x, y, z) R 3 0 : ax + by + cz = t } has no interior lattice points for 0 < t < a+b+c, so that Ehrhart-Macdonald Reciprocity implies that L (t) = 0 for (a + b + c) < t < 0, which is equivalent to the Reciprocity Law for Dedekind-Rademacher sums. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 14

34 Even more Dedekind sums The Ehrhart quasi-polynomial for := { x R d 0 : a 1x a d x d = 1 } gives rise to the Fourier-Dedekind sum (MB Robins 2003) σ n (a 1,..., a d ; a 0 ) := 1 a 0 λ a 0=1 λ n (1 λ a 1 ) (1 λ a d ). Enumerating integer-points in polytopes: applications to number theory Matthias Beck 15

35 Even more Dedekind sums The Ehrhart quasi-polynomial for := { x R d 0 : a 1x a d x d = 1 } gives rise to the Fourier-Dedekind sum (MB Robins 2003) σ n (a 1,..., a d ; a 0 ) := 1 a 0 λ a 0=1 λ n (1 λ a 1 ) (1 λ a d ). The identity L (0) = 1 implies the Reciprocity Law for Zagier s higherdimensional Dedekind sums, whereas L (t) = 0 for (a a d ) < t < 0 gives a new reciprocity relation which is a higher-dimensional analog of that for the the Dedekind-Rademacher sum. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 15

36 Partition functions and the Frobenius problem The Ehrhart quasi-polynomial L (t) = # { (m 1,..., m d ) Z d 0 : m 1 a m d a d = t } is the restricted partition function p A (t) for A = {a 1,..., a d } Enumerating integer-points in polytopes: applications to number theory Matthias Beck 16

37 Partition functions and the Frobenius problem The Ehrhart quasi-polynomial L (t) = # { (m 1,..., m d ) Z d 0 : m 1 a m d a d = t } is the restricted partition function p A (t) for A = {a 1,..., a d } Frobenius problem: find the largest value for t such that p A (t) = 0 Enumerating integer-points in polytopes: applications to number theory Matthias Beck 16

38 Partition functions and the Frobenius problem The Ehrhart quasi-polynomial L (t) = # { (m 1,..., m d ) Z d 0 : m 1 a m d a d = t } is the restricted partition function p A (t) for A = {a 1,..., a d } Frobenius problem: find the largest value for t such that p A (t) = 0 Polytopal corollaries: p A ( t) = ( 1) d 1 p A (t (a a d )) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 16

39 Partition functions and the Frobenius problem The Ehrhart quasi-polynomial L (t) = # { (m 1,..., m d ) Z d 0 : m 1 a m d a d = t } is the restricted partition function p A (t) for A = {a 1,..., a d } Frobenius problem: find the largest value for t such that p A (t) = 0 Polytopal corollaries: p A ( t) = ( 1) d 1 p A (t (a a d )) Upper bounds on the Frobenius number Enumerating integer-points in polytopes: applications to number theory Matthias Beck 16

40 Partition functions and the Frobenius problem The Ehrhart quasi-polynomial L (t) = # { (m 1,..., m d ) Z d 0 : m 1 a m d a d = t } is the restricted partition function p A (t) for A = {a 1,..., a d } Frobenius problem: find the largest value for t such that p A (t) = 0 Polytopal corollaries: p A ( t) = ( 1) d 1 p A (t (a a d )) Upper bounds on the Frobenius number New approach on the Frobenius problem via Gröbner bases Enumerating integer-points in polytopes: applications to number theory Matthias Beck 16

41 Shameless plug M. Beck & S. Robins Computing the continuous discretely Integer-point enumeration in polyhedra To appear in Springer Undergraduate Texts in Mathematics Preprint available at math.sfsu.edu/beck MSRI Summer Graduate Program at Banff (August 6 20) Enumerating integer-points in polytopes: applications to number theory Matthias Beck 17

42 Coefficients and roots of Ehrhart polynomials Integral (convex) polytope P convex hull of finitely many points in Z d Then L P (t) = c d t d + + c 0 is a polynomial in t. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 18

43 Coefficients and roots of Ehrhart polynomials Integral (convex) polytope P convex hull of finitely many points in Z d Then L P (t) = c d t d + + c 0 is a polynomial in t We know (intrinsic) geometric interpretations of c d, c d 1, and c 0. What about the other coefficients? Enumerating integer-points in polytopes: applications to number theory Matthias Beck 18

44 Coefficients and roots of Ehrhart polynomials Integral (convex) polytope P convex hull of finitely many points in Z d Then L P (t) = c d t d + + c 0 is a polynomial in t We know (intrinsic) geometric interpretations of c d, c d 1, and c 0. What about the other coefficients? What can be said about the roots of Ehrhart polynomials? Enumerating integer-points in polytopes: applications to number theory Matthias Beck 18

45 Coefficients and roots of Ehrhart polynomials Integral (convex) polytope P convex hull of finitely many points in Z d Then L P (t) = c d t d + + c 0 is a polynomial in t We know (intrinsic) geometric interpretations of c d, c d 1, and c 0. What about the other coefficients? What can be said about the roots of Ehrhart polynomials? Theorem (Stanley 1980) The generating function t 0 L P(t) x t can be written in the form f(x), where f(x) is a polynomial of degree at most (1 x) d+1 d with nonnegative integer coefficients. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 18

46 Coefficients and roots of Ehrhart polynomials Integral (convex) polytope P convex hull of finitely many points in Z d Then L P (t) = c d t d + + c 0 is a polynomial in t We know (intrinsic) geometric interpretations of c d, c d 1, and c 0. What about the other coefficients? What can be said about the roots of Ehrhart polynomials? Theorem (Stanley 1980) The generating function t 0 L P(t) x t can be written in the form f(x), where f(x) is a polynomial of degree at most (1 x) d+1 d with nonnegative integer coefficients. The inequalities f(x) 0 and c d 1 > 0 are currently the sharpest constraints on Ehrhart coefficients. Are there others? Enumerating integer-points in polytopes: applications to number theory Matthias Beck 18

47 Roots of Ehrhart polynomials are special Easy fact: L P has no integer roots besides d, d + 1,..., 1. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 19

48 Roots of Ehrhart polynomials are special Easy fact: L P has no integer roots besides d, d + 1,..., 1. Theorem (MB DeLoera Develin Pfeifle Stanley 2005) (1) The roots of Ehrhart polynomials of lattice d-polytopes are bounded in norm by 1 + (d + 1)!. (2) All real roots are in [ d, d/2 ). (3) For any positive real number r there exist an Ehrhart polynomial of sufficiently large degree with a real root strictly larger than r. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 19

49 Roots of Ehrhart polynomials are special Easy fact: L P has no integer roots besides d, d + 1,..., 1. Theorem (MB DeLoera Develin Pfeifle Stanley 2005) (1) The roots of Ehrhart polynomials of lattice d-polytopes are bounded in norm by 1 + (d + 1)!. (2) All real roots are in [ d, d/2 ). (3) For any positive real number r there exist an Ehrhart polynomial of sufficiently large degree with a real root strictly larger than r. Open problems: Improve the bound in (1). The upper bound in (2) is not sharp, for example, it can be improved to 1 for dim P = 4. Can one obtain a better (general) upper bound? Enumerating integer-points in polytopes: applications to number theory Matthias Beck 19

50 Roots of Ehrhart polynomials are special Easy fact: L P has no integer roots besides d, d + 1,..., 1. Theorem (MB DeLoera Develin Pfeifle Stanley 2005) (1) The roots of Ehrhart polynomials of lattice d-polytopes are bounded in norm by 1 + (d + 1)!. (2) All real roots are in [ d, d/2 ). (3) For any positive real number r there exist an Ehrhart polynomial of sufficiently large degree with a real root strictly larger than r. Open problems: Improve the bound in (1). The upper bound in (2) is not sharp, for example, it can be improved to 1 for dim P = 4. Can one obtain a better (general) upper bound? Conjecture: All roots α satisfy d Re α d 1. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 19

51 Roots of some tetrahedra Enumerating integer-points in polytopes: applications to number theory 0 Matthias Beck 20

52 Roots of the Birkhoff polytopes Enumerating integer-points in polytopes: applications to number theory Matthias Beck 21

53 (Weak) semimagic squares revisited H n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t Enumerating integer-points in polytopes: applications to number theory Matthias Beck 22

54 (Weak) semimagic squares revisited H n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t Theorem (MB Pixton 2003) H n (t) = 1 (2πi) n m 1 + +m n =n ( (z 1 z n ) t 1 n m 1,..., m n ) n k=1 ( z t+n 1 k j k (z k z j ) ) mk dz where denotes that we only sum over those n-tuples of non-negative integers satisfying m m n = n and m m r > r for 1 r < n Enumerating integer-points in polytopes: applications to number theory Matthias Beck 22

55 (Weak) semimagic squares revisited H n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t Theorem (MB Pixton 2003) H n (t) = 1 (2πi) n m 1 + +m n =n ( (z 1 z n ) t 1 n m 1,..., m n ) n k=1 ( z t+n 1 k j k (z k z j ) ) mk dz where denotes that we only sum over those n-tuples of non-negative integers satisfying m m n = n and m m r > r for 1 r < n Computation of H n for n 9 and vol B n for n 10 Enumerating integer-points in polytopes: applications to number theory Matthias Beck 22

56 Birkhoff volumes n vol B n / / / / Enumerating integer-points in polytopes: applications to number theory Matthias Beck 23

57 Weak magic squares M n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t j x jj = t j x j,n j = t is an (Ehrhart) quasi-polynomial in t (MB Cohen Cuomo Gribelyuk 2003) For n 3, deg M n = n 2 2n 1 Enumerating integer-points in polytopes: applications to number theory Matthias Beck 24

58 Weak magic squares M n (t) := # x 11 x 1n.. x n1... x nn Z n2 0 : j x jk = t k x jk = t j x jj = t j x j,n j = t is an (Ehrhart) quasi-polynomial in t (MB Cohen Cuomo Gribelyuk 2003) For n 3, deg M n = n 2 2n 1 Open problem: What is the period of M n? Is it always > 1 for n 2? Enumerating integer-points in polytopes: applications to number theory Matthias Beck 24

59 Strong magic squares M n(t) # magic n n-squares with distinct entries and magic sum t Enumerating integer-points in polytopes: applications to number theory Matthias Beck 25

60 Strong magic squares M n(t) # magic n n-squares with distinct entries and magic sum t Theorem (MB Zaslavsky 2006) M n(t) is the Ehrhart quasi-polynomial of an inside-out polytope, satisfying Mn(t) = µ (ˆ0, u ) L u R n 2 (t), >0 u L where L is the intersection lattice of the hyperplane arrangement { xi = x j : 1 i < j n 2} and µ is its Möbius function. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 25

61 For example... 2t 2 32t = 2 9 (t2 16t + 72) if t 0 mod 18, 2t 2 32t+78 9 = 2 9 (t 3)(t 13) if t 3 mod 18, M (t) = 2t 2 32t = 2 9 (t 6)(t 10) if t 6 mod 18, 2t 2 32t = 2 9 (t 7)(t 9) if t 9 mod 18, 2t 2 32t+96 9 = 2 9 (t 4)(t 12) if t 12 mod 18, 2t 2 32t = 2 9 (t2 16t + 51) if t 15 mod 18, 0 if t 0 mod 3. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 26

62 Magic dice (a Monthly problem) Given a 3 3-square, we form three 3-sided dice, as follows: the sides of die i are labelled with the numbers in row i. We say die i beats die j if we expect die i to show a bigger number than die j more than half the time. (a) Suppose the square is a (strong) magic square whose entries are 1, 2,..., 9. Prove that no die beats the other two and no die loses to the other two. Every die beats one die and loses to the other die. (b) Show the same is true for any strong magic square. (c) Suppose the square is a semimagic square whose entries are 1, 2,..., 9. Show the same conclusion holds as in (a) and (b). (d) But, there are semimagic squares for which one die beats both other dice. Enumerating integer-points in polytopes: applications to number theory Matthias Beck 27