FROM STURM, SYLVESTER, WITT AND WALL TO THE PRESENT DAY Andrew - PowerPoint PPT Presentation

1 FROM STURM, SYLVESTER, WITT AND WALL TO THE PRESENT DAY Andrew Ranicki http://www.maths.ed.ac.uk/˜aar/ University of Edinburgh Oxford, 28th April, 2016

2 Introduction ◮ In 1829 Sturm proved a theorem calculating the number of real roots of a non-zero real polynomial P ( X ) ∈ R [ X ] in an interval [ a , b ] ⊂ R , using the Euclidean algorithm in R [ X ] and counting sign changes. ◮ In 1853 Sylvester interpreted Sturm’s theorem using continued fractions and the signature of a tridiagonal quadratic form. In fact, this was the first application of the signature! ◮ The survey paper of ´ Etienne Ghys and A.R. http://arxiv.org/abs/1512.09258 Signatures in algebra, topology and dynamics includes a modern interpretation of the results of Sturm and Sylvester in terms of the “Witt group” of quadratic forms over the function field R ( X ). ◮ History, algebra, topology – and even some number theory!

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

4 Sturm’s problem ◮ Major problem in early 19th century How many real roots of a degree n real polynomial P ( X ) ∈ R [ X ] are there in an interval [ a , b ] ⊂ R ? ◮ Sturm’s 1829 formula for the numbers of roots involved the Sturm sequences : the remainders and quotients in the Euclidean algorithm (with sign change) in R [ X ] for finding the greatest common divisor of P 0 ( X ) = P ( X ), 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 real roots of P 0 ( X ), P 1 ( X ) , . . . , P n ( X ) are distinct and non-zero, so that deg( P k ( X )) = n − k and P n ( X ) is a non-zero constant.

5 Variation ◮ The variation of p = ( p 0 , p 1 , . . . , p n ) ∈ ( R \{ 0 } ) n +1 is the number of sign changes p 0 → p 1 → · · · → p n , which 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 functions 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 var( P ∗ ( x )) = var( P 0 ( x ) , P 1 ( x ) , . . . , P n ( x )) ∈ { 0 , 1 , . . . , n } is defined.

6 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 .

7 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 .

8 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 • • •

9 James Joseph Sylvester (1814-1897)

10 Sylvester’s 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)

11 The signature ◮ Definition The signature of a symmetric n × n matrix S = ( s ij ) 1 � i , j � n is τ ( S ) = τ + ( S ) − τ − ( S ) ∈ {− n , − n + 1 , . . . , n − 1 , n } with τ + ( S ) (resp. τ − ( S )) the number of positive (resp. negative) eigenvalues. ◮ Law of Inertia (Sylvester (1852)) For any invertible n × n matrix A = ( a ij ) with transpose A ∗ = ( a ji ) τ ( A ∗ SA ) = τ ( S ) . ◮ Theorem (Sylvester (1853), Jacobi (1857), Gundelfinger (1881), Frobenius (1895)) The signature of a symmetric n × n matrix S in R with the principal minors µ k = µ k ( S ) = det( s ij ) 1 � i , j � k non-zero is ∑ n sign( µ k /µ k − 1 ) = n − 2 var( µ ) . τ ( S ) = k =1

12 The tridiagonal symmetric matrix (Jacobi, Sylvester) ◮ Definition The tridiagonal symmetric matrix of q = ( q 1 , q 2 , . . . , q n ) is   1 0 . . . 0 q 1   1 q 2 1 . . . 0     0 1 . . . 0 q 3 Tri( q ) =     . . . . ... . . . .   . . . . 0 0 0 . . . q n ◮ Tridiagonal Signature Theorem For q ∈ R n the signature of Tri( q ) is ∑ n sign( µ k /µ k − 1 ) = n − 2 var( µ ) τ (Tri( q )) = k =1 assuming µ k = µ k (Tri( q )) = det(Tri( q 1 , q 2 , . . . , q k )) ̸ = 0 ∈ R .

13 Continued fractions and the Sturm functions ◮ 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 P k ( X ) [ Q k +1 ( X ) , Q k +2 ( X ) , . . . , Q n ( X )] = P k +1 ( X ) ∈ R ( X ) (0 � k � n − 1) P k ( X ) / P n ( X ) = det(Tri( Q k +1 ( X ) , Q k +2 ( X ) , . . . , Q n ( X ))).

14 Sylvester’s Duality Theorem (1853) ◮ 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 the minors of Tri( Q 1 ( X ) , Q 2 ( X ) , . . . , Q n ( X )) 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 ] . ◮ Sylvester’s Duality Theorem Let x ∈ R be regular for P ( X ). The variations of the sequence of the numerators of the convergents and reverse convergents are equal var( P 0 ( x ) , P 1 ( x ) , . . . , P n ( x )) = var( P ∗ 0 ( x ) , P ∗ 1 ( x ) , . . . , P ∗ n ( x )) .

15 Sylvester’s reformulation of Sturm’s Theorem ◮ Theorem (S.-S.) The number of real roots of P ( X ) ∈ R [ X ] in an interval [ a , b ] with regular a < b can be calculated from the signatures of the tridiagonal symmetric matrices Tri( Q ∗ ( x )) for x = a and b var( P 0 ( a ) , P 1 ( a ) , . . . , P n ( a )) − var( P 0 ( b ) , P 1 ( b ) , . . . , P n ( b )) ) = ( τ (Tri( Q ∗ ( b ))) − τ (Tri( Q ∗ ( a ))) / 2 ∈ { 0 , 1 , 2 , . . . , n } . ◮ Proof For any regular x ∈ [ a , b ] var( P 0 ( x ) , P 1 ( x ) , . . . , P n ( x )) = var( P ∗ 0 ( x ) , P ∗ 1 ( x ) , . . . , P ∗ n ( x )) (by the Duality Theorem) ( ) = n − τ (Tri( Q ∗ ( x )) / 2 ∈ { 0 , 1 , 2 , . . . , n } .