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The partial-fractions method for counting solutions to integral linear systems Matthias Beck, MSRI www.msri.org/people/members/matthias/ arXiv: math.CO/0309332

Vector partition functions A – an ( m × d ) -integral matrix b ∈ Z m � x ∈ Z d � Goal: Compute vector partition function φ A ( b ) := # ≥ 0 : A x = b (defined for b in the nonnegative linear span of the columns of A ) The partial-fractions method for counting solutions to integral linear systems Matthias Beck 2

Vector partition functions A – an ( m × d ) -integral matrix b ∈ Z m � x ∈ Z d � Goal: Compute vector partition function φ A ( b ) := # ≥ 0 : A x = b (defined for b in the nonnegative linear span of the columns of A ) Applications in... ◮ Number Theory (partitions) ◮ Discrete Geometry (polyhedra) ◮ Commutative Algebra (Hilbert series) ◮ Algebraic Geometry (toric varieties) ◮ Representation Theory (tensor product multiplicities) ◮ Optimization (integer programming) ◮ Chemistry, Biology, Physics, Computer Science, Economics... The partial-fractions method for counting solutions to integral linear systems Matthias Beck 2

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # The partial-fractions method for counting solutions to integral linear systems Matthias Beck 3

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � � x ∈ R d Translate & introduce slack variables − → P = ≥ 0 : A x = b � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # The partial-fractions method for counting solutions to integral linear systems Matthias Beck 3

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � � x ∈ R d Translate & introduce slack variables − → P = ≥ 0 : A x = b � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # = φ A ( t b ) (for fixed b ) The partial-fractions method for counting solutions to integral linear systems Matthias Beck 3

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � x ∈ R d � Translate & introduce slack variables − → P = ≥ 0 : A x = b � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # = φ A ( t b ) (for fixed b ) 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 the functions 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. Theorem (Ehrhart, Macdonald 1970) L P ( − t ) = ( − 1) dim P L P ◦ ( t ) The partial-fractions method for counting solutions to integral linear systems Matthias Beck 3

Vector partition theorems � x ∈ Z d � φ A ( b ) := # ≥ 0 : A x = b n c n ( b ) b n with coefficients c n that are Quasi-polynomial – a finite sum � functions of b which are periodic in every component of b . A matrix is unimodular if every square submatrix has determinant ± 1 . Theorem (Sturmfels 1995) φ A ( b ) is a piecewise-defined quasi-polynomial The regions of R m in which φ A ( b ) is a in b of degree d − rank( A ) . single quasi-polynomial are polyhedral. If A is unimodular then φ A is a piecewise-defined polynomial. B 2002) Let r k denote the sum of the entries in the k th row of Theorem (M A , and let r = ( r 1 , . . . , r m ) . Then φ A ( b ) = ( − 1) d − rank A φ A ( − b − r ) The partial-fractions method for counting solutions to integral linear systems Matthias Beck 4

Issues... ◮ Compute the regions of (quasi-)polynomiality of φ A ( b ) ◮ Given one such region, compute the (quasi-)polynomial φ A ( b ) φ A ( t b ) z t can be computed in polynomial time � ◮ Barvinok: t ≥ 0 The partial-fractions method for counting solutions to integral linear systems Matthias Beck 5

Euler’s generating function   | | | � x ∈ Z d � φ A ( b ) := # ≥ 0 : A x = b A = · · · c 1 c 2 c d   | | | φ A ( b ) equals the coefficient of z b := z b 1 1 · · · z b m m of the function 1 (1 − z c 1 ) · · · (1 − z c d ) expanded as a power series centered at z = 0 . The partial-fractions method for counting solutions to integral linear systems Matthias Beck 6

Euler’s generating function   | | | � x ∈ Z d � φ A ( b ) := # ≥ 0 : A x = b A = · · · c 1 c 2 c d   | | | φ A ( b ) equals the coefficient of z b := z b 1 1 · · · z b m m of the function 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. The partial-fractions method for counting solutions to integral linear systems Matthias Beck 6

Euler’s generating function   | | | � x ∈ Z d � φ A ( b ) := # ≥ 0 : A x = b A = · · · c 1 c 2 c d   | | | φ A ( b ) equals the coefficient of z b := z b 1 1 · · · z b m m of the function 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 φ A ( b ) = const (1 − z c 1 ) · · · (1 − z c d ) z b The partial-fractions method for counting solutions to integral linear systems Matthias Beck 6

Partial fractions 1 φ A ( b ) = const (1 − z c 1 ) · · · (1 − z c d ) z b Expand into partial fractions in z 1 :   b 1 d 1 1 A k ( z , b 1 ) B j ( z ) � � (1 − z c 1 ) · · · (1 − z c d ) z b = 1 − z c k +   z b 2 z j 2 · · · z b m m 1 j =1 k =1 Here A k and B j are polynomials in z 1 , rational functions in z 2 , . . . , z m , and exponential in b 1 . The partial-fractions method for counting solutions to integral linear systems Matthias Beck 7

Partial fractions 1 φ A ( b ) = const (1 − z c 1 ) · · · (1 − z c d ) z b Expand into partial fractions in z 1 :   b 1 d 1 1 A k ( z , b 1 ) B j ( z ) � � (1 − z c 1 ) · · · (1 − z c d ) z b = 1 − z c k +   z b 2 z j 2 · · · z b m m 1 j =1 k =1 Here A k and B j are polynomials in z 1 , rational functions in z 2 , . . . , z m , and exponential in b 1 . d 1 A k ( z , b 1 ) � φ A ( b ) = const z 2 ,...,z m const z 1 z b 2 2 · · · z b m 1 − z c k m k =1 The partial-fractions method for counting solutions to integral linear systems Matthias Beck 7

Partial fractions 1 φ A ( b ) = const (1 − z c 1 ) · · · (1 − z c d ) z b Expand into partial fractions in z 1 :   b 1 d 1 1 A k ( z , b 1 ) B j ( z ) � � (1 − z c 1 ) · · · (1 − z c d ) z b = 1 − z c k +   z b 2 z j 2 · · · z b m m 1 k =1 j =1 Here A k and B j are polynomials in z 1 , rational functions in z 2 , . . . , z m , and exponential in b 1 . d 1 A k ( z , b 1 ) � φ A ( b ) = const z 2 ,...,z m const z 1 z b 2 2 · · · z b m 1 − z c k m k =1 d 1 A k (0 , z 2 , . . . , z m , b 1 ) � = const z b 2 2 · · · z b m 1 − (0 , z 2 , . . . , z m ) c k m k =1 The partial-fractions method for counting solutions to integral linear systems Matthias Beck 7

Advantages ◮ easy to implement ◮ allows symbolic computation ◮ constraints which define the regions of (quasi-)polynomiality are obtained “automatically” The partial-fractions method for counting solutions to integral linear systems Matthias Beck 8

An example ❅ ❅ ❅ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ x 1 , x 2 , x 3 , x 4 ≥ 0 ❅ ❅ ❅ x 1 + 2 x 2 + x 3 = a ❅ ❅ x 1 + x 2 + x 4 = b ❅ ❅ ❅ ❅ 1 φ A ( a, b ) = const (1 − zw )(1 − z 2 w )(1 − z )(1 − w ) z a w b The partial-fractions method for counting solutions to integral linear systems Matthias Beck 9

An example ❅ ❅ ❅ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ x 1 , x 2 , x 3 , x 4 ≥ 0 ❅ ❅ ❅ x 1 + 2 x 2 + x 3 = a ❅ ❅ x 1 + x 2 + x 4 = b ❅ ❅ ❅ ❅ 1 φ A ( a, b ) = const (1 − zw )(1 − z 2 w )(1 − z )(1 − w ) z a w b z b +1 z 2 b +3 1 b 1 . . . (1 − z ) 2 (1 − z )(1 − z 2 ) (1 − z )(1 − z 2 ) � (1 − zw )(1 − z 2 w )(1 − w ) w b = − 1 − zw + 1 − z 2 w + + w k 1 − w k =1 The partial-fractions method for counting solutions to integral linear systems Matthias Beck 9

An example ❅ ❅ ❅ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ x 1 , x 2 , x 3 , x 4 ≥ 0 ❅ ❅ ❅ x 1 + 2 x 2 + x 3 = a ❅ ❅ x 1 + x 2 + x 4 = b ❅ ❅ ❅ ❅ 1 φ A ( a, b ) = const (1 − zw )(1 − z 2 w )(1 − z )(1 − w ) z a w b z b +1 z 2 b +3 1 b 1 . . . (1 − z ) 2 (1 − z )(1 − z 2 ) (1 − z )(1 − z 2 ) � (1 − zw )(1 − z 2 w )(1 − w ) w b = − 1 − zw + 1 − z 2 w + + w k 1 − w k =1 z b +1 z 2 b +3 1 � 1 � φ A ( a, b ) = const − (1 − z ) 2 + (1 − z )(1 − z 2 ) + (1 − z )(1 − z 2 ) (1 − z ) z a − z b − a +1 z 2 b − a +3 � � 1 = const (1 − z ) 3 + (1 − z ) 2 (1 − z 2 ) + (1 − z ) 2 (1 − z 2 ) z a The partial-fractions method for counting solutions to integral linear systems Matthias Beck 9