Periods Numerical computation and applications Pierre Lairez Inria - PowerPoint PPT Presentation

Periods Numerical computation and applications Pierre Lairez Inria Saclay Séminaire de lancement ANR « De rerum natura » 24 février 2020, Palaiseau

What is a period? A period is the integral on a closed path of a rational function in one or several variables with rational coefficients. “Rational coefficients” may mean • coefficients in Q • coefficients in C ( t ) , the period is a function of t . 1

What is a period? A period is the integral on a closed path of a rational function in one or several variables with rational coefficients. “Rational coefficients” may mean • coefficients in Q • coefficients in C ( t ) , the period is a function of t . Etymology 1

What is a period? A period is the integral on a closed path of a rational function in one or several variables with rational coefficients. “Rational coefficients” may mean • coefficients in Q • coefficients in C ( t ) , the period is a function of t . Etymology • 2 π is a period of the sine. 1

What is a period? A period is the integral on a closed path of a rational function in one or several variables with rational coefficients. “Rational coefficients” may mean • coefficients in Q • coefficients in C ( t ) , the period is a function of t . Etymology • 2 π is a period of the sine. � z d x • arcsin( z ) = � 1 − x 2 0 1

What is a period? A period is the integral on a closed path of a rational function in one or several variables with rational coefficients. “Rational coefficients” may mean • coefficients in Q • coefficients in C ( t ) , the period is a function of t . Etymology • 2 π is a period of the sine. � z d x • arcsin( z ) = � 1 − x 2 0 � d x • 2 π = � • • 1 − x 2 ∞ − 1 1 1

What is a period? A period is the integral on a closed path of a rational function in one or several variables with rational coefficients. “Rational coefficients” may mean • coefficients in Q • coefficients in C ( t ) , the period is a function of t . Etymology • 2 π is a period of the sine. � z d x • arcsin( z ) = � 1 − x 2 0 � � 1 − x 2 = 1 d x d x d y • 2 π = � • • y 2 − (1 − x 2 ) π i ∞ − 1 1 1

Periods with a parameter Complete elliptic integral 2

Periods with a parameter Complete elliptic integral 1 An ellipse eccentricity t t F ′ O F major radius 1 perimeter E ( t ) 2

Periods with a parameter Complete elliptic integral 1 An ellipse eccentricity t t F ′ O F major radius 1 perimeter E ( t ) � � 1 1 − t 2 x 2 E ( t ) = 2 1 − x 2 d x − 1 2

Periods with a parameter Complete elliptic integral 1 An ellipse eccentricity t t F ′ O F major radius 1 perimeter E ( t ) � � 1 − t 2 x 2 • • • • E ( t ) = 1 − x 2 d x − 1 1 ∞ 2

Periods with a parameter Complete elliptic integral 1 An ellipse eccentricity t t F ′ O F major radius 1 perimeter E ( t ) � � 1 − t 2 x 2 • • • • E ( t ) = 1 − x 2 d x − 1 1 ∞ Euler (1733) ( t − t 3 ) E ′′ + (1 − t 2 ) E ′ + tE = 0 2

Periods with a parameter Complete elliptic integral 1 An ellipse eccentricity t t F ′ O F major radius 1 perimeter E ( t ) � � 1 − t 2 x 2 • • • • E ( t ) = 1 − x 2 d x − 1 1 ∞ Euler (1733) ( t − t 3 ) E ′′ + (1 − t 2 ) E ′ + tE = 0 Liouville (1834) Not expressible in terms of elementary functions 2

Periods with a parameter Complete elliptic integral 1 An ellipse eccentricity t t F ′ O F major radius 1 perimeter E ( t ) � � 1 − t 2 x 2 • • • • E ( t ) = 1 − x 2 d x − 1 1 ∞ Euler (1733) ( t − t 3 ) E ′′ + (1 − t 2 ) E ′ + tE = 0 Liouville (1834) Not expressible in terms of elementary functions since then Many applications in algebraic geometry geometry of the cycles ↔ analytic properties of the periods 2

Content Computing periods with a parameter Volume of semialgebraic sets Picard rank of K3 surfaces Perpectives

Computing periods with a parameter

Differential equations as a data structure I 4

Differential equations as a data structure I Representation of algebraic numbers 4

Differential equations as a data structure I Representation of algebraic numbers � � � � explicit (also 3 ) 5 + 2 6 2 + 4

Differential equations as a data structure I Representation of algebraic numbers � � � � explicit (also 3 ) 5 + 2 6 2 + implicit x 4 − 10 x 2 + 1 = 0 (+ root location) 4

Differential equations as a data structure I Representation of algebraic numbers � � � � explicit (also 3 ) 5 + 2 6 2 + implicit x 4 − 10 x 2 + 1 = 0 (+ root location) 4

Differential equations as a data structure I Representation of algebraic numbers � � � � explicit (also 3 ) 5 + 2 6 2 + implicit x 4 − 10 x 2 + 1 = 0 (+ root location) Representation of D-finite functions An example by Bostan, Chyzak, van Hoeij, and Pech (2011) 4

Differential equations as a data structure I Representation of algebraic numbers � � � � explicit (also 3 ) 5 + 2 6 2 + implicit x 4 − 10 x 2 + 1 = 0 (+ root location) Representation of D-finite functions An example by Bostan, Chyzak, van Hoeij, and Pech (2011) � � � � 27 w (2 − 3 w ) 1/3 2/3 � 2 F 1 � t 2 (1 − 4 w ) 3 explicit 1 + 6 · d w (1 − 4 w )(1 − 64 w ) 0 4

Differential equations as a data structure I Representation of algebraic numbers � � � � explicit (also 3 ) 5 + 2 6 2 + implicit x 4 − 10 x 2 + 1 = 0 (+ root location) Representation of D-finite functions An example by Bostan, Chyzak, van Hoeij, and Pech (2011) � � � � 27 w (2 − 3 w ) 1/3 2/3 � 2 F 1 � t 2 (1 − 4 w ) 3 explicit 1 + 6 · d w (1 − 4 w )(1 − 64 w ) 0 implicit t ( t − 1)(64 t − 1)(3 t − 2)(6 t + 1) y ′′′ + (4608 t 4 − 6372 t 3 + 813 t 2 + 514 t − 4) y ′′ + 4(576 t 3 − 801 t 2 − 108 t + 74) y ′ = 0 (+ init. cond.) 4

Differential equations as a data structure II What can we compute? 5

Differential equations as a data structure II What can we compute? • addition, multiplication, composition with algebraic functions 5

Differential equations as a data structure II What can we compute? • addition, multiplication, composition with algebraic functions • power series expansion 5

Differential equations as a data structure II What can we compute? • addition, multiplication, composition with algebraic functions • power series expansion • equality testing , given differential equations and initial condtions 5

Differential equations as a data structure II What can we compute? • addition, multiplication, composition with algebraic functions • power series expansion • equality testing , given differential equations and initial condtions • numerical analytic continuation with certified precision (D. V. Chudnovsky and G. V. Chudnovsky 1990; van der Hoeven 1999; Mezzarobba 2010) More on this later. 5

The Picard-Fuchs equation Back to the periods R ( t , x 1 ,..., x n ) a rational function 6

The Picard-Fuchs equation Back to the periods R ( t , x 1 ,..., x n ) a rational function γ ⊂ C n a n -cycle ( n -dim. compact submanifold) which avoids the poles of R , for t ∈ U ⊂ C 6

The Picard-Fuchs equation Back to the periods R ( t , x 1 ,..., x n ) a rational function γ ⊂ C n a n -cycle ( n -dim. compact submanifold) which avoids the poles of R , for t ∈ U ⊂ C � define y ( t ) � R ( t , x 1 ,..., x n )d x 1 ··· d x n , for t ∈ U γ 6

The Picard-Fuchs equation Back to the periods R ( t , x 1 ,..., x n ) a rational function γ ⊂ C n a n -cycle ( n -dim. compact submanifold) which avoids the poles of R , for t ∈ U ⊂ C � define y ( t ) � R ( t , x 1 ,..., x n )d x 1 ··· d x n , for t ∈ U γ wanted a differential equation a r ( t ) y ( r ) +···+ a 1 ( t ) y ′ + a 0 ( t ) y = 0 , with polynomial coefficients 6

The Picard-Fuchs equation Back to the periods R ( t , x 1 ,..., x n ) a rational function γ ⊂ C n a n -cycle ( n -dim. compact submanifold) which avoids the poles of R , for t ∈ U ⊂ C � define y ( t ) � R ( t , x 1 ,..., x n )d x 1 ··· d x n , for t ∈ U γ wanted a differential equation a r ( t ) y ( r ) +···+ a 1 ( t ) y ′ + a 0 ( t ) y = 0 , with polynomial coefficients One equation fits all cycles, the Picard-Fuchs equation . 6

A computational handle Perimeter of an ellipse R ( t , x , y ) �� � �� � � 1 − t 2 x 2 1 1 recall E ( t ) = 1 − x 2 d x = d x d y 1 − t 2 x 2 2 π i 1 − ( 1 − x 2 ) y 2 Picard-Fuchs equation ( t − t 3 ) E ′′ + (1 − t 2 ) E ′ + tE = 0 7

A computational handle Perimeter of an ellipse R ( t , x , y ) �� � �� � � 1 − t 2 x 2 1 1 recall E ( t ) = 1 − x 2 d x = d x d y 1 − t 2 x 2 2 π i 1 − ( 1 − x 2 ) y 2 Picard-Fuchs equation ( t − t 3 ) E ′′ + (1 − t 2 ) E ′ + tE = 0 Computational proof ( t − t 3 ) ∂ 2 R ∂ t 2 + (1 − t 2 ) ∂ R ∂ t + tR = � 2 t ( − 1 + t 2 ) x ( 1 + x 3 ) y 3 � � � − t ( − 1 − x + x 2 + x 3 ) y 2 ( − 3 + 2 x + y 2 + x 2 ( − 2 + 3 t 2 − y 2 )) ∂ + ∂ 2 2 ∂ x ∂ y ( − 1 + y 2 + x 2 ( t 2 − y 2 )) ( − 1 + y 2 + x 2 ( t 2 − y 2 )) 7