Computing the continuous discretely: The magic quest for a volume Matthias Beck San Francisco State University math.sfsu.edu/beck
Joint work with... Dennis Pixton (Birkhoff volume) ◮ Ricardo Diaz and Sinai Robins (Fourier-Dedekind sums) ◮ Ira Gessel and Takao Komatsu (restricted partition function) ◮ Jesus De Loera, Mike Develin, Julian Pfeifle, Richard Stanley (roots of ◮ Ehrhart polynomials) Integer-point enumeration in polytopes Matthias Beck 2
Birkhoff polytope x 11 · · · x 1 n � j x jk = 1 for all 1 ≤ k ≤ n ∈ R n 2 . . . . B n = . . ≥ 0 : � k x jk = 1 for all 1 ≤ j ≤ n x n 1 . . . x nn Integer-point enumeration in polytopes Matthias Beck 3
Birkhoff polytope x 11 · · · x 1 n � j x jk = 1 for all 1 ≤ k ≤ n ∈ R n 2 . . . . B n = . . ≥ 0 : � k x jk = 1 for all 1 ≤ j ≤ n x n 1 . . . x nn ◮ B n is a convex polytope of dimension ( n − 1) 2 ◮ Vertices are the n × n -permutation matrices. Integer-point enumeration in polytopes Matthias Beck 3
Birkhoff polytope x 11 · · · x 1 n � j x jk = 1 for all 1 ≤ k ≤ n ∈ R n 2 . . . . B n = . . ≥ 0 : � k x jk = 1 for all 1 ≤ j ≤ n x n 1 . . . x nn ◮ B n is a convex polytope of dimension ( n − 1) 2 ◮ Vertices are the n × n -permutation matrices. vol B n =? Integer-point enumeration in polytopes Matthias Beck 3
Birkhoff polytope x 11 · · · x 1 n � j x jk = 1 for all 1 ≤ k ≤ n ∈ R n 2 . . . . B n = . . ≥ 0 : � k x jk = 1 for all 1 ≤ j ≤ n x n 1 . . . x nn ◮ B n is a convex polytope of dimension ( n − 1) 2 ◮ Vertices are the n × n -permutation matrices. vol B n =? � tX ∩ Z d � # One approach: for X ⊂ R d , vol X = lim t d t →∞ Integer-point enumeration in polytopes Matthias Beck 3
(Weak) semimagic squares � t B n ∩ Z n 2 � H n ( t ) := # x 11 · · · x 1 n � j x jk = t . . ∈ Z n 2 . . = # . . ≥ 0 : � k x jk = t x n 1 . . . x nn Integer-point enumeration in polytopes Matthias Beck 4
(Weak) semimagic squares � t B n ∩ Z n 2 � H n ( t ) := # x 11 · · · x 1 n � j x jk = t . . ∈ Z n 2 . . = # . . ≥ 0 : � k x jk = t x n 1 . . . x nn Theorem (Ehrhart, Stanley 1973, conjectured by Anand-Dumir-Gupta 1966) H n ( t ) is a polynomial in t of degree ( n − 1) 2 . Integer-point enumeration in polytopes Matthias Beck 4
(Weak) semimagic squares � t B n ∩ Z n 2 � H n ( t ) := # x 11 · · · x 1 n � j x jk = t . . ∈ Z n 2 . . = # . . ≥ 0 : � k x jk = t x n 1 . . . x nn Theorem (Ehrhart, Stanley 1973, conjectured by Anand-Dumir-Gupta 1966) H n ( t ) is a polynomial in t of degree ( n − 1) 2 . For example... ◮ H 1 ( t ) = 1 ◮ H 2 ( t ) = t + 1 8 t 4 + 3 4 t 3 + 15 8 t 2 + 9 � t +3 � t +2 = 1 � � ◮ (MacMahon 1905) H 3 ( t ) = 3 + 4 t + 1 4 2 Integer-point enumeration in polytopes Matthias Beck 4
Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � � x ∈ R d � Alternative description: P = ≥ 0 : A x = b ⇄ Integer-point enumeration in polytopes Matthias Beck 5
Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � � x ∈ R d � Alternative description: P = ≥ 0 : A x = b ⇄ � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # Integer-point enumeration in polytopes Matthias Beck 5
Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � � x ∈ R d � Alternative description: P = ≥ 0 : A x = b ⇄ � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # Quasi-polynomial – c d ( t ) t d + c d − 1 ( t ) t d − 1 + · · · + c 0 ( t ) where c k ( t ) are periodic Theorem (Ehrhart 1967) If P is a rational polytope, then... ◮ L P ( t ) and L P ◦ ( t ) are quasi-polynomials in t of degree dim P ◮ If P has integer vertices, then L P and L P ◦ are polynomials ◮ Leading term: vol( P ) (suitably normalized) ◮ L P (0) = χ ( P ) ◮ (Macdonald 1970) L P ( − t ) = ( − 1) dim P L P ◦ ( t ) Integer-point enumeration in polytopes Matthias Beck 5
(Weak) semimagic squares revisited H n ( t ) = L B n ( t ) x 11 · · · x 1 n � j x jk = t . . ∈ Z n 2 . . = # . . ≥ 0 : � k x jk = t x n 1 . . . x nn x 11 · · · x 1 n � j x jk = t . . ∈ Z n 2 . . L B ◦ n ( t ) = # . . > 0 : � k x jk = t x n 1 . . . x nn Integer-point enumeration in polytopes Matthias Beck 6
(Weak) semimagic squares revisited H n ( t ) = L B n ( t ) x 11 · · · x 1 n � j x jk = t . . ∈ Z n 2 . . . . = # ≥ 0 : � k x jk = t x n 1 . . . x nn x 11 · · · x 1 n � j x jk = t . . ∈ Z n 2 . . L B ◦ n ( t ) = # . . > 0 : � k x jk = t x n 1 . . . x nn L B ◦ n ( t ) = L B n ( t − n ) , so by Ehrhart-Macdonald reciprocity (Ehrhart, Stanley 1973) H n ( − n − t ) = ( − 1) ( n − 1) 2 H n ( t ) H n ( − 1) = · · · = H n ( − n + 1) = 0 . Integer-point enumeration in polytopes Matthias Beck 6
Computation of Ehrhart (quasi-)polynomials Pommersheim (1993): 3-dimensional tetrahedra – connection to ◮ Dedekind sum b − 1 cot πka cot πk � b b k =1 Integer-point enumeration in polytopes Matthias Beck 7
Computation of Ehrhart (quasi-)polynomials Pommersheim (1993): 3-dimensional tetrahedra – connection to ◮ Dedekind sum b − 1 cot πka cot πk � b b k =1 t ≥ 0 L P ( t ) x t is polynomial-time Barvinok (1993): In fixed dimension, � ◮ computable Integer-point enumeration in polytopes Matthias Beck 7
Computation of Ehrhart (quasi-)polynomials Pommersheim (1993): 3-dimensional tetrahedra – connection to ◮ Dedekind sum b − 1 cot πka cot πk � b b k =1 t ≥ 0 L P ( t ) x t is polynomial-time Barvinok (1993): In fixed dimension, � ◮ computable Formulas by Danilov, Brion-Vergne, Kantor-Khovanskii-Puklikov, Diaz- ◮ Robins, Chen, Baldoni-DeLoera-Szenes-Vergne, Lasserre-Zeron, . . . Integer-point enumeration in polytopes Matthias Beck 7
Euler’s generating function | | | � x ∈ R d � P := ≥ 0 : A x = b A = · · · c 1 c 2 c d | | | Integer-point enumeration in polytopes Matthias Beck 8
Euler’s generating function | | | � x ∈ R d � P := ≥ 0 : A x = b A = · · · c 1 c 2 c d | | | L P ( t ) equals the coefficient of z t b := z tb 1 · · · z tb m of the function 1 m 1 (1 − z c 1 ) · · · (1 − z c d ) expanded as a power series centered at z = 0 . Integer-point enumeration in polytopes Matthias Beck 8
Euler’s generating function | | | � x ∈ R d � P := ≥ 0 : A x = b A = · · · c 1 c 2 c d | | | L P ( t ) equals the coefficient of z t b := z tb 1 · · · z tb m of the function 1 m 1 (1 − z c 1 ) · · · (1 − z c d ) expanded as a power series centered at z = 0 . Proof Expand each factor into a geometric series. Integer-point enumeration in polytopes Matthias Beck 8
Euler’s generating function | | | � x ∈ R d � P := ≥ 0 : A x = b A = · · · c 1 c 2 c d | | | L P ( t ) equals the coefficient of z t b := z tb 1 · · · z tb m of the function 1 m 1 (1 − z c 1 ) · · · (1 − z c d ) expanded as a power series centered at z = 0 . Proof Expand each factor into a geometric series. Equivalently, 1 L P ( t ) = const (1 − z c 1 ) · · · (1 − z c d ) z t b Integer-point enumeration in polytopes Matthias Beck 8
Partition functions and the Frobenius problem Restricted partition function for A = { a 1 , . . . , a d } ( m 1 , . . . , m d ) ∈ Z d � � p A ( t ) = # ≥ 0 : m 1 a 1 + · · · + m d a d = t Integer-point enumeration in polytopes Matthias Beck 9
Partition functions and the Frobenius problem Restricted partition function for A = { a 1 , . . . , a d } ( m 1 , . . . , m d ) ∈ Z d � � p A ( t ) = # ≥ 0 : m 1 a 1 + · · · + m d a d = t Frobenius problem: find the largest value for t such that p A ( t ) = 0 Integer-point enumeration in polytopes Matthias Beck 9
Partition functions and the Frobenius problem Restricted partition function for A = { a 1 , . . . , a d } ( m 1 , . . . , m d ) ∈ Z d � � p A ( t ) = # ≥ 0 : m 1 a 1 + · · · + m d a d = t Frobenius problem: find the largest value for t such that p A ( t ) = 0 p A ( t ) = L P ( t ) where ( x 1 , . . . , x d ) ∈ R d � � P = ≥ 0 : x 1 a 1 + · · · + x d a d = 1 Integer-point enumeration in polytopes Matthias Beck 9
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