The Bellows Theorem (Proof) Giovanni Viglietta JAIST July 5, 2018 - PowerPoint PPT Presentation

The Bellows Theorem (Proof) Giovanni Viglietta JAIST – July 5, 2018 The Bellows Theorem (Proof)

Bellows theorem: statement Theorem (Sabitov, 1996) The volume V of a polyhedron (of any genus) with edge lengths ℓ 1 , · · · , ℓ e satisfies V N + A N − 1 V N − 1 + · · · + A 2 V 2 + A 1 V + A 0 = 0 , where the coefficients A i are polynomials in Q [ ℓ 2 1 , · · · , ℓ 2 e ] and only depend on the combinatorial structure of the polyhedron. The Bellows Theorem (Proof)

Bellows theorem: statement Theorem (Sabitov, 1996) The volume V of a polyhedron (of any genus) with edge lengths ℓ 1 , · · · , ℓ e satisfies V N + A N − 1 V N − 1 + · · · + A 2 V 2 + A 1 V + A 0 = 0 , where the coefficients A i are polynomials in Q [ ℓ 2 1 , · · · , ℓ 2 e ] and only depend on the combinatorial structure of the polyhedron. As the polyhedron flexes maintaining its edge lengths ℓ i fixed, the coefficients A i remain the same. Hence the volume V is a root of the same polynomial, and it can only take finitely many values. Corollary (Bellows theorem) The volume of a polyhedron is constant throughout any flexing. Note: for the sake of the bellows theorem, it is not restrictive to consider only polyhedra with triangular faces. The Bellows Theorem (Proof)

Proof roadmap Frobenius The sum of algebraic Cayley-Menger companion matrix numbers is algebraic determinant Kronecker product Bellows theorem Elimination theory Surgery of polyhedra Classification of Sylvester matrix closed surfaces The Bellows Theorem (Proof)

Proof roadmap Frobenius The sum of algebraic Cayley-Menger companion matrix numbers is algebraic determinant Kronecker product Bellows theorem Elimination theory Surgery of polyhedra Classification of Sylvester matrix closed surfaces The Bellows Theorem (Proof)

Characteristic polynomial of a matrix The characteristic polynomial of an n × n matrix A is the monic degree- n polynomial c A ( x ) = det( x I − A ) . Example: � 2 � 1 The characteristic polynomial of the matrix is − 1 0 � 2 � � 1 0 � 1 �� � x − 2 − 1 � = x 2 − 2 x + 1 det x · − = det 0 1 − 1 0 1 x Lemma The roots of c A ( x ) are precisely the eigenvalues of A . The Bellows Theorem (Proof)

Frobenius companion matrix of a polynomial The Frobenius companion matrix of the monic polynomial P ( x ) = x n + a n − 1 x n − 1 + · · · + a 1 x + a 0 is the n × n matrix:   0 0 · · · 0 − a 0 1 0 · · · 0 − a 1     0 1 · · · 0 − a 2 F P =    . . . .  ... . . . .   . . . .   0 0 · · · 1 − a n − 1 The Bellows Theorem (Proof)

Frobenius companion matrix of a polynomial Lemma The eigenvalues of F P are precisely the roots of P ( x ) . Let us prove by induction on n that c F P ( x ) = P ( x ) .  x 0 · · · 0 a 0  − 1 x · · · 0 a 1   c F P ( x ) = det( x I − F P ) = det  = . . . . ...  . . . .  . . . .  0 0 · · · − 1 x + a n − 1  x 0 · · · 0 a 1   − 1 x 0 · · · 0  − 1 x · · · 0 a 2 0 − 1 x · · · 0      . . . .   . . . .  ... ... +( − 1) n +1 a 0 · det . . . . . . . . x · det     . . . . . . . .         0 0 · · · x a n − 2 0 0 0 · · · x     0 0 · · · − 1 x + a n − 1 0 0 0 · · · − 1 = x · ( x n − 1 + a n − 1 x n − 2 + · · · + a 2 x + a 1 ) + ( − 1) n +1 a 0 · ( − 1) n − 1 = x n + a n − 1 x n − 1 + · · · + a 1 x + a 0 = P ( x ) The Bellows Theorem (Proof)

Kronecker product If A is an m × n matrix and B is a p × q matrix, the Kronecker product A ⊗ B is the mp × nq matrix:   a · · · a B B 11 1 n . . . . . . A ⊗ B =   . . .   a · · · a B B m 1 mn Example: � 5 � � 5 �  6 6   5 6 10 12  1 · 2 · � 1 � � 5 � 2 6 7 8 7 8 7 8 14 16     ⊗ =  =     3 4 7 8 � 5 6 � � 5 6 � 15 18 20 24    3 · 4 · 7 8 7 8 21 24 28 32 The Bellows Theorem (Proof)

Kronecker product: mixed-product property If the matrix products AC and BD are well defined, then:     a 11 B . . . a 1 n B c 11 D . . . c 1 p D . . . . ... ... . . . . ( A ⊗ B )( C ⊗ D ) =     . . . .     a m 1 B . . . a mn B c n 1 D . . . c np D � n � n   k =1 a 1 k c k 1 BD . . . k =1 a 1 k c kp BD . . ... . . =  = AC ⊗ BD   . .  � n � n k =1 a mk c k 1 BD . . . k =1 a mk c kp BD The Bellows Theorem (Proof)

The sum of polynomial roots is a polynomial root Lemma If A and B are monic polynomials with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] , there is a monic polynomial C with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] such that, if A ( α ) = 0 and B ( β ) = 0 , then C ( α + β ) = 0 . The Bellows Theorem (Proof)

The sum of polynomial roots is a polynomial root Lemma If A and B are monic polynomials with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] , there is a monic polynomial C with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] such that, if A ( α ) = 0 and B ( β ) = 0 , then C ( α + β ) = 0 . Let A and B be the Frobenius companion matrices of A and B . Then there are vectors x and y such that Ax = α x and By = β y . The Bellows Theorem (Proof)

The sum of polynomial roots is a polynomial root Lemma If A and B are monic polynomials with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] , there is a monic polynomial C with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] such that, if A ( α ) = 0 and B ( β ) = 0 , then C ( α + β ) = 0 . Let A and B be the Frobenius companion matrices of A and B . Then there are vectors x and y such that Ax = α x and By = β y . ( A ⊗ I + I ⊗ B )( x ⊗ y ) = ( A ⊗ I )( x ⊗ y ) + ( I ⊗ B )( x ⊗ y ) (apply the mixed-product property) = ( Ax ⊗ Iy )+( Ix ⊗ By ) = ( α x ⊗ y )+( x ⊗ β y ) = ( α + β )( x ⊗ y ) Hence α + β is an eigenvalue of the matrix A ⊗ I + I ⊗ B , and therefore α + β is a root of its characteristic polynomial C . The Bellows Theorem (Proof)

The sum of polynomial roots is a polynomial root Lemma If A and B are monic polynomials with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] , there is a monic polynomial C with coefficients in Q [ ℓ 2 1 , · · · , ℓ 2 e ] such that, if A ( α ) = 0 and B ( β ) = 0 , then C ( α + β ) = 0 . Let A and B be the Frobenius companion matrices of A and B . Then there are vectors x and y such that Ax = α x and By = β y . ( A ⊗ I + I ⊗ B )( x ⊗ y ) = ( A ⊗ I )( x ⊗ y ) + ( I ⊗ B )( x ⊗ y ) (apply the mixed-product property) = ( Ax ⊗ Iy )+( Ix ⊗ By ) = ( α x ⊗ y )+( x ⊗ β y ) = ( α + β )( x ⊗ y ) Hence α + β is an eigenvalue of the matrix A ⊗ I + I ⊗ B , and therefore α + β is a root of its characteristic polynomial C . The coefficients of C were obtained by adding and multiplying coefficients of A and B , and thus they are in Q [ ℓ 2 1 , · · · , ℓ 2 e ] . The Bellows Theorem (Proof)

Proof roadmap Frobenius The sum of algebraic Cayley-Menger companion matrix numbers is algebraic determinant Kronecker product Bellows theorem Elimination theory Surgery of polyhedra Classification of Sylvester matrix closed surfaces The Bellows Theorem (Proof)

Proof roadmap Frobenius The sum of algebraic Cayley-Menger companion matrix numbers is algebraic determinant Kronecker product Bellows theorem Elimination theory Surgery of polyhedra Classification of Sylvester matrix closed surfaces The Bellows Theorem (Proof)

Cayley-Menger determinant Lemma If x 1 , x 2 , x 3 , x 4 , x 5 ∈ R 3 and d ij = � x i − x j � , then d 2 d 2 d 2 d 2   0 1 12 13 14 15 d 2 d 2 d 2 d 2 0 1 21 23 24 25   d 2 d 2 d 2 d 2 0 1   det 31 32 34 35 = 0 .  d 2 d 2 d 2 d 2  0 1 41 42 43 45   d 2 d 2 d 2 d 2  0 1  51 52 53 54 1 1 1 1 1 0 The Bellows Theorem (Proof)