Combinatorics of simple polytopes and differential equations Buchstaber, Victor M. 2008 MIMS EPrint: 2008.54 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester http://eprints.maths.manchester.ac.uk/ ISSN 1749-9097 Reports available from: And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK
Combinatorics of simple polytopes and differential equations. Victor M Buchstaber Steklov Institute, RAS, Moscow � b uchstab@mi.ras.ru � School of Mathematics, University of Manchester � V ictor.Buchstaber@manchester.ac.uk � Manchester 21 February 2008
Abstract Simple polytopes play important role in applications of algebraic geometry to physics. They are also main objects in toric topology. There is a commutative associative ring P generated by simple polytopes. The ring P possesses a natural derivation d , which comes from the boundary operator. We shall describe a ring homomorphism from the ring P to the ring of polynomials Z [ t , α ] transforming the operator d to the partial derivative ∂ / ∂ t . This result opens way to a relation between polytopes and differential equations. As it has turned out, certain important series of polytopes (including some recently discovered) lead to fundamental non-linear differential equations in partial derivatives. 1
Definition . A polytope P n of dimension n is said to be simple if every vertex of P is the intersection of exactly n facets, i.e. faces of dimension n − 1 . Definition . Two polytopes P 1 and P 2 of the same dimension are said to be combinatorially equivalent if there is a bijection between their sets of faces that preserves the inclusion relation. Definition . A combinatorial polytope is a class of combinatorial equivalent geometrical polytopes. The collection of all n -dimensional combinatorial simple polytopes is denoted by P n . 2
An Abelian group structure on P n is induced by the disjoint union of polytopes. The zero element of the group P n is the empty set. The weak direct sum � P = P n n � 0 yields a graded commutative associative ring. The product P n 1 P m of homogeneous elements P n 1 and 2 P m 2 is given by the direct product P n 1 × P m 2 . The unit element is a single point. Remarks: 1. The direct product P n 1 × P m 2 of simple polytopes P n 1 and P m 2 is a simple polytope as well. 2. Each face of a simple polytope is again a simple polytope. 3
Let P n ∈ P n be a simple polytope. Denote by dP n ∈ P n − 1 the disjoint union of all its facets. Lemma . We have a linear operator of degree − 1 d : P − → P , such that d ( P n 1 P m 2 ) = ( dP n 1 ) P m 2 + P n 1 ( dP m 2 ). Examples: d ∆ n = ( n + 1) ∆ n − 1 , dI n = n ( dI ) I n − 1 = 2 nI n − 1 , where ∆ n is the standard n -simplex and I n = I ×· · ·× I is the standard n -cube. 4
Face-polynomial. Consider the linear map F : P − → Z [ t , α ], which send a simple polytope P n to the homogeneous face-polynomial F ( P n ) = α n + f n − 1,1 α n − 1 t + · · · + f 1, n − 1 α t n − 1 + f 0, n t n , where f k , n − k = f k , n − k ( P n ) is the number of its k -dimensional faces. Thus, f n − 1,1 is the number of facets and f 0, n is the number of vertex. Note that f k , n − k = f n − k − 1 , where f ( P n ) = ( f 0 , . . . , f n − 1 ) is f -vector of P n . Theorem The mapping F is a ring homomorphism such that F ( dP n ) = ∂ ∂ tF ( P n ). 5
Corollary. F ( I n ) = ( α + 2 t ) n , F ( ∆ n ) = ( α + t ) n +1 − t n +1 . α Set � F ( I n ) x n +1 . U ( t , x ; α , I ) = n � 0 Lemma . The function U ( t , x ; α , I ) is the solution of the equation ∂ tU ( t , x ) = 2 x 2 ∂ ∂ ∂ xU ( t , x ) x with the initial condition U (0, x ) = 1 − α x . We have x U ( t , x ; α , I ) = 1 − ( α + 2 t ) x . 6
Set � F ( ∆ n ) x n +2 . U ( t , x ; α , ∆ ) = n � 0 Lemma . The function U ( t , x ; α , ∆ ) is the solution of the equation ∂ tU ( t , x ) = x 2 ∂ ∂ ∂ xU ( t , x ) x 2 with the initial condition U (0, x ) = 1 − α x . We have x 2 U ( t , x ; α , ∆ ) = (1 − tx )(1 − ( α + t ) x ). 7
Consider the series of Stasheff polytopes (the associahedra) As = { As n = K n +2 , n � 0 } . Each facet of As n is As i × As j , i � 0, i + j = n − 1 , where embedding µ k : As i × As j → ∂ As n , 1 � k � i +2 , correspondes to the pairing ( a 1 · · · a i +2 ) × ( b 1 · · · b j +2 ) − → − → a 1 · · · a k − 1 ( b 1 · · · b j +2 ) a k +1 · · · a i +2 . Lemma . i +2 � � � dAs n = µ k ( As i × As j ) = ( i +2)( As i × As j ). i + j = n − 1 k =1 i + j = n − 1 Corollary. � ∂ ∂ tF ( As n ) = ( i + 2) F ( As i ) F ( As j ). i + j = n − 1 8
Set � F ( As n ) x n +2 . U ( t , x ; α , As ) = n � 0 Theorem . The function U ( t , x ; α , As ) is the solution of the Hopf equation ∂ tU ( t , x ) = U ( t , x ) ∂ ∂ ∂ xU ( t , x ) x 2 with the initial condition U (0, x ) = 1 − α x . The function U ( t , x ; α , As ) satisfies the equation t ( α + t ) U 2 − (1 − ( α + 2 t ) x ) U + x 2 = 0. 9
Quasilinear Burgers–Hopf Equation The Hopf equation (Eberhard F.Hopf, 1902–1983) is the equation U t + f ( U ) U x = 0. The Hopf equation with f ( U ) = U is a limit case of the following equations: U t + UU x = µ U xx (the Burgers equation), U t + UU x = ε U xxx (the Korteweg–de Vries equation). The Burgers equation (Johannes M.Burgers, 1895–1981) occurs in various areas of applied mathematics (fluid and gas dynamics, acoustics, traffic flow). It used for describing of wave processes with velocity u and viscosity coefficient µ . The case µ = 0 is a prototype of equations whose solution can develop discontinuities (shock waves). K-d-V equation (Diederik J.Korteweg, 1848–1941 and Hugo M. de Vries, 1848–1935) was introduced as equation for the long waves over water (in 1895). It appears also in plasma physics. Today K-d-V equation is a most famous equation in soliton theory. 10
Let us consider the Burgers equation U t = UU x − µ U xx . U = U 0 + � µ k U k . Then Set k � 1 � � � µ k U k , t = µ k U k µ k U k , x U 0 + U 0, x + − U 0, t + k � 1 k � 1 k � 1 � µ k +1 U k , xx . − µ U 0, xx − k � 1 Thus we obtain: U 0, t = U 0 U 0, x , U 1, t = ( U 0 U 1 ) x − U 0, xx . 11
For simple polytopes, the formula for the Euler characteristic admits a generalization in the form of Dehn–Sommerville relations. In terms of the f -vector of an n -dimensional polytope P , they can be written as follows: n ( − 1) n − j � j � � f k − 1 = f j − 1 , k = 0, 1, . . . , n . k j = k Consider the ring homomorphism T : Z [ t , α ] − → Z [ t , α ], T p ( t , α ) = p ( t + α , − α ). Theorem . The Dehn–Sommerville relations are equivalent to the formula T F ( P n ) = F ( P n ). 12
Consider the ring homomorphism → Z [ z , α ] : λ ( t ) = 1 λ : Z [ t , α ] − 2( z − α ), λ ( α ) = α , and � T ( z ) = z , � � T : Z [ z , α ] − → Z [ z , α ] : T ( α ) = − α . Lemma . � T λ p ( t , α ) = λ Tp ( t , α ) Corollary . For any P n ∈ P n the polynomial p ( z , α ) = λ F ( P n ) is such that p ( z , α ) = p ( z , − α ). Examples . Set additionally λ ( x ) = x . Then x 1. λ U ( t , x ; α , I ) = 1 − zx . x 2 � . 2. λ U ( t , x ; α , ∆ ) = � �� 1 − 1 1 − 1 2 ( z − α ) x 2 ( z + α ) x 3. Set U = U ( t , x ; α , As ) . The function � U = λ U satisfies the equation U 2 − 4(1 − zx ) � U + 4 x 2 = 0. ( z − α )( z + α ) � 13
The solution of this quadratic equation with the initial x 2 � U (0, x ) = condition 1 − α x gives � (1 − zx ) − (1 − 2 zx + α 2 x 2 ) 1/2 � ( z 2 − α 2 ) � U = 2 . Consider two vectors r , r ′ such that | r | = 1, | r ′ | = α x , � r , r ′ � = zx . Then | r || r ′ | cos( r , r ′ ) = α x cos( r , r ′ ) = zx . z 2 − α 2 = − α 2 sin 2 ( r , r ′ ) , Thus, z = α cos( r , r ′ ), 1 − zx = | r | 2 − � r , r ′ � = � r , r − r ′ � , (1 − 2 zx + α 2 x 2 ) 1/2 = | r − r ′ | . Lemma . The function � U satisfies the equation � � α 2 sin 2 ( r , r ′ ) � | r − r ′ | − � r , r − r ′ � U = 2 . 14
We have � � � � d x ( z 2 − α 2 ) � U = 2 − x + = | r − r ′ | dz � z � ∞ � α n L n x n , = 2 x α n � 1 where L n ( · ) are Legendre polynomials. We have � z � � � z �� 1 d ( z 2 − α 2 ) d = . L n dzL n n ( n + 1) α dz α Thus, � z � α n � U = 2 ∂ x n +1 � , n ( n + 1) L n ∂ z α n � 1 � z � ∂ 2 � � ∂ x 2 = 2 ∂ U x n − 1 α n L n . ∂ z α n � 1 x ∂ 2 ∂ x 2 U = ∂ 1 | r − r ′ | . Corollary . ∂ t 15
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