Properties of Laurent coefficients of multivariate rational - PowerPoint PPT Presentation

Properties of Laurent coefficients of multivariate rational functions Workshop on Computer Algebra in Combinatorics Erwin Schroedinger Institute Armin Straub November 14, 2017 University of South Alabama includes joint work with and Frits Beukers Marc Houben Wadim Zudilin (Utrecht University) (Utrecht University) (University of Newcastle/ Radboud Universiteit) Properties of Laurent coefficients of multivariate rational functions Armin Straub 1 / 34

Goal of this talk We introduce and advertise two questions about rational functions like 1 � A ( n ) x n . = 1 − ( x 1 + x 2 + x 3 ) + 4 x 1 x 2 x 3 n ∈ Z 3 � 0 Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

Goal of this talk We introduce and advertise two questions about rational functions like 1 � A ( n ) x n . = 1 − ( x 1 + x 2 + x 3 ) + 4 x 1 x 2 x 3 n ∈ Z 3 � 0 When has a rational function the Gauss property ? Q That is, when do the following congruences hold? A ( n p r ) ≡ A ( n p r − 1 ) (mod p r ) Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

Goal of this talk We introduce and advertise two questions about rational functions like 1 � A ( n ) x n . = 1 − ( x 1 + x 2 + x 3 ) + 4 x 1 x 2 x 3 n ∈ Z 3 � 0 When has a rational function the Gauss property ? Q That is, when do the following congruences hold? A ( n p r ) ≡ A ( n p r − 1 ) (mod p r ) When is a rational function positive ? Q That is, when is A ( n ) > 0 for all n ? Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

Goal of this talk We introduce and advertise two questions about rational functions like 1 � A ( n ) x n . = 1 − ( x 1 + x 2 + x 3 ) + 4 x 1 x 2 x 3 n ∈ Z 3 � 0 When has a rational function the Gauss property ? Q That is, when do the following congruences hold? A ( n p r ) ≡ A ( n p r − 1 ) (mod p r ) When is a rational function positive ? Q That is, when is A ( n ) > 0 for all n ? In both cases, we will wonder about an explicit characterization. These are not conjectures because our evidence is limited. Computer algebra! Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

Goal of this talk We introduce and advertise two questions about rational functions like 1 � A ( n ) x n . = 1 − ( x 1 + x 2 + x 3 ) + 4 x 1 x 2 x 3 n ∈ Z 3 � 0 Here, the diagonal coefficients are the Franel numbers EG n � 3 � n � A ( n, n, n ) = . k k =0 • As seen in previous talks, simple multivariate generating functions can be enormously useful, for instance, in computing asymptotics. • Time permitting, more on Ap´ ery-like numbers later. . . Properties of Laurent coefficients of multivariate rational functions Armin Straub 2 / 34

I Gauss congruences Properties of Laurent coefficients of multivariate rational functions Armin Straub 3 / 34

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. THM a φ ( m ) ≡ 1 (mod m ) Euler if a is coprime to m . Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. THM a φ ( m ) ≡ 1 (mod m ) Euler if a is coprime to m . THM d ) a d ≡ 0 � µ ( m (mod m ) Gauss d | m obius function: µ ( n ) = ( − 1) # of p | n if n is square-free, µ ( n ) = 0 else M¨ Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

The classical Gauss congruence THM a p ≡ a (mod p ) Fermat if p is prime. THM a φ ( m ) ≡ 1 (mod m ) Euler if a is coprime to m . THM d ) a d ≡ 0 � µ ( m (mod m ) Gauss d | m obius function: µ ( n ) = ( − 1) # of p | n if n is square-free, µ ( n ) = 0 else M¨ If m = p r then only d = p r , d = p r − 1 contribute, and we get EG a p r ≡ a p r − 1 (mod p r ) . Properties of Laurent coefficients of multivariate rational functions Armin Straub 4 / 34

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG L n +1 = L n + L n − 1 • a ( n ) = L n Lucas numbers : L 0 = 2 , L 1 = 1 Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG L n +1 = L n + L n − 1 • a ( n ) = L n Lucas numbers : L 0 = 2 , L 1 = 1 n � n �� n + k � � • a ( n ) = D n Delannoy numbers : D n = k k k =0 Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

Gauss congruences DEF a ( n ) satisfies the Gauss congruences if, for all primes p , a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) . � µ ( m Equivalently, d ) a ( d ) ≡ 0 (mod m ) . d | m • a ( n ) = a n EG L n +1 = L n + L n − 1 • a ( n ) = L n Lucas numbers : L 0 = 2 , L 1 = 1 n � n �� n + k � � • a ( n ) = D n Delannoy numbers : D n = k k k =0 • Later, we allow a ( n ) ∈ Q . If the Gauss congruences hold for all but finitely many p , we say that the sequence (or its GF) has the Gauss property . • Similarly, for multivariate sequences a ( n ) , we require a ( m p r ) ≡ a ( m p r − 1 ) (mod p r ) . Properties of Laurent coefficients of multivariate rational functions Armin Straub 5 / 34

More sequences satisfying Gauss congruences a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) (G) • realizable sequences a ( n ) , i.e., for some map T : X → X , a ( n ) = # { x ∈ X : T n x = x } “points of period n ” Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05 In fact, up to a positivity condition, (G) characterizes realizability. Properties of Laurent coefficients of multivariate rational functions Armin Straub 6 / 34

More sequences satisfying Gauss congruences a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) (G) • realizable sequences a ( n ) , i.e., for some map T : X → X , a ( n ) = # { x ∈ X : T n x = x } “points of period n ” Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05 In fact, up to a positivity condition, (G) characterizes realizability. • a ( n ) = trace( M n ) J¨ anichen ’21, Schur ’37; also: Arnold, Zarelua where M is an integer matrix Properties of Laurent coefficients of multivariate rational functions Armin Straub 6 / 34

More sequences satisfying Gauss congruences a ( mp r ) ≡ a ( mp r − 1 ) (mod p r ) (G) • realizable sequences a ( n ) , i.e., for some map T : X → X , a ( n ) = # { x ∈ X : T n x = x } “points of period n ” Everest–van der Poorten–Puri–Ward ’02, Arias de Reyna ’05 In fact, up to a positivity condition, (G) characterizes realizability. • a ( n ) = trace( M n ) J¨ anichen ’21, Schur ’37; also: Arnold, Zarelua where M is an integer matrix � ∞ � a ( n ) • (G) is equivalent to exp � n T n ∈ Z [[ T ]] . n =1 This is a natural condition in formal group theory . Properties of Laurent coefficients of multivariate rational functions Armin Straub 6 / 34

Minton’s theorem THM f ∈ Q ( x ) has the Gauss property if and only if f is a Q -linear Minton, combination of functions xu ′ ( x ) /u ( x ) , with u ∈ Z [ x ] . 2014 Properties of Laurent coefficients of multivariate rational functions Armin Straub 7 / 34

Minton’s theorem THM f ∈ Q ( x ) has the Gauss property if and only if f is a Q -linear Minton, combination of functions xu ′ ( x ) /u ( x ) , with u ∈ Z [ x ] . 2014 • If u ( x ) = � s i =1 (1 − α i x ) then s s xu ′ ( x ) α i x 1 � � u ( x ) = − 1 − α i x = s − 1 − α i x. i =1 i =1 Properties of Laurent coefficients of multivariate rational functions Armin Straub 7 / 34

Minton’s theorem THM f ∈ Q ( x ) has the Gauss property if and only if f is a Q -linear Minton, combination of functions xu ′ ( x ) /u ( x ) , with u ∈ Z [ x ] . 2014 • If u ( x ) = � s i =1 (1 − α i x ) then s s xu ′ ( x ) α i x 1 � � u ( x ) = − 1 − α i x = s − 1 − α i x. i =1 i =1 • Assuming the α i are distinct, � s s � 1 x n = � � � � α n trace( M n ) x n , 1 − α i x = i i =1 n � 0 i =1 n � 0 where M is the companion matrix of � s i =1 ( x − α i ) = x s u (1 /x ) . Properties of Laurent coefficients of multivariate rational functions Armin Straub 7 / 34