POLYNOMIALS, QUADRATIC FORMS AND THE TOPOLOGY OF MANIFOLDS For Erik - PowerPoint PPT Presentation

1 POLYNOMIALS, QUADRATIC FORMS AND THE TOPOLOGY OF MANIFOLDS For Erik Kjær Pedersen Andrew Ranicki http://www.maths.ed.ac.uk/˜aar/ University of Edinburgh Copenhagen, 20th June, 2016

2 In Edinburgh

3 “Signatures, braids and Seifert surfaces” ◮ A collection of old and new papers to appear later in 2016 in a volume edited by ´ Etienne Ghys and myself of the Brazilian online journal Ensaios Matem´ aticos: ◮ ´ Etienne Ghys and Andrew Ranicki Signatures in algebra, topology and dynamics ◮ Jean-Marc Gambaudo and ´ Etienne Ghys Braids and signatures ◮ Arjeh Cohen and Jack van Wijk Visualization of Seifert Surfaces ◮ Julia Collins An algorithm for computing the Seifert matrix of a link from a braid representation ◮ Maxime Bourrigan Quasimorphismes sur les groupes de tresses et forme de Blanchfield ◮ Chris Palmer Seifert matrices of braids with applications to isotopy and signatures

4 In the beginning ◮ Major problem from early 19th century How many real roots does a degree n real polynomial P ( X ) ∈ R [ X ] have in an interval [ a , b ] ⊂ R ? That is, calculate # R -roots( P ( X ); [ a , b ]) = |{ x ∈ [ a , b ] | P ( x ) = 0 }| ∈ { 0 , 1 . . . , n } ◮ In 1829 Sturm solved the problem algorithmically, using the Euclidean algorithm in R [ X ] for the greatest common divisor of P ( X ) and P ′ ( X ) and counting sign changes. ◮ In 1853 Sylvester interpreted Sturm’s theorem using the continued fraction expansion of P ( X ) / P ′ ( X ) and the signatures of symmetric matrices. This was the first ever application of the signature! ◮ There have been very many applications of the signatures since then, particularly in the topology of manifolds.

5 Plan for today 1. The Sturm algorithm for # R -roots( P ( X ); [ a , b ]) for a degree n real polynomial P ( X ) ∈ R [ X ]. 2. The Sylvester expression for # R -roots( P ( X ); [ a , b ]) as a difference of Witt classes ( ) ( R n , Tri( b )) − ( R n , Tri( a )) / 2 ∈ W ( R ) = Z (signature) of tridiagonal symmetric matrices (= forms) over R . 3. The Ghys-R. expression for # R -roots( P ( X ); [ a , b ]) in terms of the Witt class ⊕ ⊕ ( R ( X ) , P ( X )) ∈ W ( R ( X )) = Z ⊕ Z 2 (multisignature) ∞ ∞ with R ( X ) the field of fractions of the polynomial ring R [ X ]. 4. Tridiagonal symmetric matrices in the Milnor-Hirzebruch plumbing of sphere bundles, and the work of Barge-Lannes on the Maslov index and Bott periodicity.

6 Jacques Charles Fran¸ cois Sturm (1803-1855)

7 The Sturm sequences ◮ Sturm’s 1829 algorithmic formula for the number of real roots involved the Sturm sequences of P ( X ) ∈ R [ X ]: the remainders P k ( X ) and quotients Q k ( X ) in the Euclidean algorithm (with sign change) in R [ X ] for finding the greatest common divisor of P 0 ( X ) = P ( X ) and P 1 ( X ) = P ′ ( X ) P ∗ ( X ) = ( P 0 ( X ) , . . . , P n ( X )) , Q ∗ ( X ) = ( Q 1 ( X ) , . . . , Q n ( X )) with deg( P k +1 ( X )) < deg( P k ( X )) � n − k and P k − 1 ( X ) + P k +1 ( X ) = P k ( X ) Q k ( X ) (1 � k � n ) . ◮ Simplifying assumption P ( X ) is generic: the roots of P 0 ( X ), P 1 ( X ) , . . . , P n ( X ) are distinct, so that deg( P k ( X )) = n − k , P n ( X ) is a non-zero constant, and deg( Q k ( X )) = 1.

8 Variation ◮ The variation var( p ) of p = ( p 0 , p 1 , . . . , p n ) ∈ ( R \{ 0 } ) n +1 is the number of sign changes p 0 → p 1 → · · · → p n . ◮ The variation is expressed in terms of the sign changes p k − 1 → p k by ∑ n var( p ) = ( n − sign( p k / p k − 1 )) / 2 ∈ { 0 , 1 , . . . , n } . k =1 ◮ Sturm’s root-counting formula involved the variations of the Sturm remainders P k ( X ) evaluated at ‘regular’ x ∈ R . ◮ Call x ∈ R regular if P k ( x ) ̸ = 0 (0 � k � n − 1), so that the variation in the values of the Sturm remainders var( P ∗ ( x )) = var( P 0 ( x ) , P 1 ( x ) , . . . , P n ( x )) ∈ { 0 , 1 , . . . , n } is defined.

9 Sturm’s Theorem I. ◮ Theorem (1829) The number of real roots of a generic P ( X ) ∈ R [ X ] in [ a , b ] ⊂ R for regular a < b is |{ x ∈ [ a , b ] | P ( x ) = 0 ∈ R }| = var( P ∗ ( a )) − var( P ∗ ( b )) . ◮ Idea of proof The function : [ a , b ] → { 0 , 1 , . . . , n } ; x �→ var( P ∗ ( a )) − var( P ∗ ( x )) f { { 1 0 jumps by 0 at root x of P k ( X ) if k = 1 , 2 , . . . , n . ◮ For k = 0 the jump in f at a root x of P 0 ( x ) is 1, since for y close to x { < 0 if y < x P 0 ( y ) P 1 ( y ) = d / dy ( P ( y ) 2 ) / 2 = > 0 if y > x , { var(+ , − ) = var( − , +) = 1 if y < x var( P 0 ( y ) , P 1 ( y )) = var(+ , +) = var( − , − ) = 0 if y > x .

10 Sturm’s Theorem II. ◮ For k = 1 , 2 , . . . , n the jump in f at a root x of P k ( x ) is 0. ◮ k = n trivial, since P n ( X ) is non-zero constant. ◮ For k = 1 , 2 , . . . , n − 1 the numbers P k − 1 ( x ), P k +1 ( x ) ̸ = 0 ∈ R have opposite signs since P k − 1 ( x ) + P k +1 ( x ) = P k ( x ) Q k ( x ) = 0 . ◮ For y , z close to x with y < x < z sign( P k − 1 ( y )) = − sign( P k +1 ( y )) = sign( P k − 1 ( z )) = − sign( P k +1 ( z )) , var( P k − 1 ( y ) , P k ( y ) , P k +1 ( y )) = var( P k − 1 ( z ) , P k ( z ) , P k +1 ( z )) = 1 , that is var(+ , + , − ) = var(+ , − , − ) = var( − , + , +) = var( − , − , +) = 1 .

11 Sturm’s theorem III. ( y, P k +1 ( y )) ( x, P k +1 ( x )) ( z, P k +1 ( z )) P k +1 • • • ( y, P k ( y )) • ( x, P k ( x )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • 0 y x z ( z, P k ( z )) • P k ( y, P k − 1 ( y )) ( x, P k − 1 ( x )) ( z, P k − 1 ( z )) P k − 1 • • •

12 James Joseph Sylvester (1814-1897)

13 Sylvester’s 4 papers related to Sturm’s theorem ◮ On the relation of Sturm’s auxiliary functions to the roots of an algebraic equation. (1841) ◮ A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. (1852) ◮ On a remarkable modification of Sturm’s Theorem (1853) ◮ On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure. (1853)

14 The signature ◮ The transpose of an n × n matrix A = ( a ij ) is A ∗ = ( a ji ). ◮ Spectral Theorem (Cauchy, 1829) For any symmetric n × n matrix S = S ∗ in R there exists an orthogonal A = A ∗− 1 with   λ 1 0 . . . 0   0 λ 2 . . . 0   A ∗ SA = diag( λ 1 , . . . , λ n ) =   . . . ... . . .   . . . 0 0 . . . λ n ◮ The signature of a symmetric n × n matrix S is ∑ n τ ( S ) = τ ( A ∗ SA ) = sign( λ i ) i =1 ◮ If S is invertible τ ( S ) = n − 2 var(1 , λ 1 , . . . , λ n ) ≡ n mod 2. ◮ Law of Inertia (Sylvester, 1853) For any invertible n × n matrix A in R τ ( S ) = τ ( A ∗ SA ) .

15 Tridiagonal symmetric matrices (Jacobi) ◮ Definition The tridiagonal symmetric matrix of q = ( q 1 , q 2 , . . . , q n ) ∈ R n is   1 0 . . . 0 q 1   1 q 2 1 . . . 0     0 1 . . . 0 q 3 Tri( q ) =     . . . . ... . . . .   . . . . 0 0 0 . . . q n ◮ The principal minors of Tri( q ) µ k = det(Tri( q 1 , q 2 , . . . , q k )) (1 � k � n ) satisfy the recurrence of the Euclidean algorithm µ k = q k µ k − 1 − µ k − 2 ( µ 0 = 1 , µ − 1 = 0) .

16 The signature of a tridiagonal matrix ◮ Theorem (Sylvester, 1853) Assume the principal minors µ k = µ k (Tri( q )) = det(Tri( q 1 , q 2 , . . . , q k )) (1 � k � n ) are non-zero. The invertible n × n matrix   ( − 1) n − 1 µ 0 /µ n − 1 − µ 0 /µ 1 1 µ 0 /µ 2 . . .   ( − 1) n − 2 µ 1 /µ n − 1 0 1 − µ 1 /µ 2 . . .     ( − 1) n − 3 µ 2 /µ n − 1 0 0 1 . . . A =     . . . . ... . . . .   . . . . 0 0 0 . . . 1 is such that A ∗ Tri( q ) A = diag( µ 1 /µ 0 , µ 2 /µ 1 , . . . , µ n /µ n − 1 ) so that n ∑ τ (Tri( q )) = sign( µ k /µ k − 1 ) = n − 2 var( µ ) . k =1

17 Continued fractions and the Sturm sequences ◮ The improper continued fraction of ( q 1 , q 2 , . . . , q n ) is 1 [ q 1 , q 2 , . . . , q n ] = q 1 − q 2 − ... − 1 q n assuming there are no divisions by 0. ◮ The continued fraction expansion of P ( X ) / P ′ ( X ) is P ( X ) / P ′ ( X ) = [ Q 1 ( X ) , Q 2 ( X ) , . . . , Q n ( X )] ∈ R ( X ) with Q 1 ( X ) , Q 2 ( X ) , . . . , Q n ( X ) the Sturm quotients. ◮ The Sturm remainders ( P 0 ( X ) , P 1 ( X ) , . . . , P n ( X )) are the numerators in the reverse convergents (0 � k � n ) [ Q k +1 ( X ) , Q k +2 ( X ) , . . . , Q n ( X )] = P k ( X ) / P k +1 ( X ) ∈ R ( X ) . ◮ P k ( X ) / P n ( X ) = det(Tri( Q k +1 ( X ) , Q k +2 ( X ) , . . . , Q n ( X )))

18 Convergents ◮ The convergents of [ Q 1 ( X ) , Q 2 ( X ) , . . . , Q n ( X )] ∈ R ( X ) are [ Q 1 ( X ) , Q 2 ( X ) , . . . , Q k ( X )] P ∗ k ( X ) = det(Tri( Q 2 ( X ) , Q 3 ( X ) , . . . , Q k ( X ))) with numerators P ∗ k ( X ) = µ k (Tri( Q 1 ( X ) , Q 2 ( X ) , . . . , Q n ( X ))) = det(Tri( Q 1 ( X ) , Q 2 ( X ) , . . . , Q k ( X ))) ∈ R [ X ] the principal minors of Tri( Q 1 ( X ) , Q 2 ( X ) , . . . , Q n ( X )).