A (truly) universal differential equation Amaury Pouly Joint work - PowerPoint PPT Presentation

Rubel’s proof in one slide − 1 1 − t 2 for − 1 < t < 1 and f ( t ) = 0 otherwise. ◮ Take f ( t ) = e ( 1 − t 2 ) 2 f ′′ ( t ) + 2 tf ′ ( t ) = 0 . It satisfies ◮ For any a , b , c ∈ R , y ( t ) = cf ( at + b ) satisfies 3 y ′ 4 y ′′ y ′′′′ 2 − 4 y ′ 4 y ′′ 2 y ′′′′ + 6 y ′ 3 y ′′ 2 y ′′′ y ′′′′ + 24 y ′ 2 y ′′ 4 y ′′′′− 12 y ′ 3 y ′′ y ′′′ 3 − 29 y ′ 2 y ′′ 3 y ′′′ 2 + 12 y ′′ 7 = 0 ◮ Can glue together arbitrary many such pieces ◮ Can arrange so that � f is solution : piecewise pseudo-linear t Conclusion : Rubel’s equation allows any piecewise pseudo-linear functions, and those are dense in C 0 13 / 24

Universal initial value problem (IVP) y 1 ( x ) x Theorem There exists a fixed (vector of) polynomial p such that for any continuous functions f and ε , there exists α ∈ R d such that y ′ ( t ) = p ( y ( t )) y ( 0 ) = α, has a unique solution y : R → R d and ∀ t ∈ R , | y 1 ( t ) − f ( t ) | � ε ( t ) . 14 / 24

Universal initial value problem (IVP) Notes : y 1 ( x ) ◮ system of ODEs, x ◮ y is analytic, ◮ we need d ≈ 300. Theorem There exists a fixed (vector of) polynomial p such that for any continuous functions f and ε , there exists α ∈ R d such that y ′ ( t ) = p ( y ( t )) y ( 0 ) = α, has a unique solution y : R → R d and ∀ t ∈ R , | y 1 ( t ) − f ( t ) | � ε ( t ) . 14 / 24

Universal initial value problem (IVP) Notes : y 1 ( x ) ◮ system of ODEs, x ◮ y is analytic, ◮ we need d ≈ 300. Theorem There exists a fixed (vector of) polynomial p such that for any continuous functions f and ε , there exists α ∈ R d such that y ′ ( t ) = p ( y ( t )) y ( 0 ) = α, has a unique solution y : R → R d and ∀ t ∈ R , | y 1 ( t ) − f ( t ) | � ε ( t ) . Remark : α is usually transcendental, but computable from f and ε 14 / 24

Universal DAE revisited y 1 ( x ) x Theorem There exists a fixed polynomial p and k ∈ N such that for any continuous functions f and ε , there exists α 0 , . . . , α k ∈ R such that p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( 0 ) = α 0 , y ′ ( 0 ) = α 1 , . . . , y ( k ) ( 0 ) = α k has a unique analytic solution and this solution satisfies such that | y ( t ) − f ( t ) | � ε ( t ) . 15 / 24

A brief stop Before I can explain the proof, you need to know more of polynomial ODEs and what I mean by programming with ODEs. 16 / 24

Generable functions : a summary Definition f : R → R is generable if ∃ d , p and y 0 such that the solution y to y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , satisfies f ( x ) = y 1 ( x ) for all x ∈ R . y 1 ( x ) x 17 / 24

Generable functions : a summary Definition f : R → R is generable if ∃ d , p and y 0 such that the solution y to y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , satisfies f ( x ) = y 1 ( x ) for all x ∈ R . y 1 ( x ) x Nice theory for the class of total and univariate generable functions : ◮ analytic ◮ contains polynomials, sin , cos , tanh , exp ◮ stable under ± , × , /, ◦ and Initial Value Problems (IVP) y ′ = f ( y ) ◮ solutions to polynomial ODEs form a very large class 17 / 24

Why is this useful? Writing polynomial ODEs by hand is hard. 18 / 24

Why is this useful? Writing polynomial ODEs by hand is hard. Using generable functions, we can build complicated multivariate partial functions using other operations, and we know they are solutions to polynomial ODEs by construction . 18 / 24

Why is this useful? Writing polynomial ODEs by hand is hard. Using generable functions, we can build complicated multivariate partial functions using other operations, and we know they are solutions to polynomial ODEs by construction . Example : almost rounding function There exists a generable function round such that for any n ∈ Z , x ∈ R , λ > 2 and µ � 0 : n − 1 2 , n + 1 then | round( x , µ, λ ) − n | � 1 ◮ if x ∈ � � 2 , 2 n − 1 2 + 1 λ , n + 1 2 − 1 then | round( x , µ, λ ) − n | � e − µ . ◮ if x ∈ � � λ See proof 18 / 24

A simplified proof binary stream digits of α generator 0 1 1 0 1 0 1 0 0 1 1 1 . . . α ∈ R ODE t E T O N This is the ideal curve, the real one is an approximation of it. 19 / 24

A simplified proof binary stream digits of α generator 0 1 1 0 1 0 1 0 0 1 1 1 . . . α ∈ R ODE t ODE t “Digital” to Analog E T Converter (fixed frequency) O N E T O Approximate Lipschitz and bounded N functions with fixed precision. That’s the trickiest part. 19 / 24

A simplified proof binary stream digits of α generator 0 1 1 0 1 0 1 0 0 1 1 1 . . . α ∈ R ODE t ODE t “Digital” to Analog Converter (fixed frequency) ODE? E T t O N We need something more : a fast-growing ODE. 19 / 24

A simplified proof binary stream digits of α generator 0 1 1 0 1 0 1 0 0 1 1 1 . . . α ∈ R ODE t ODE t “Digital” to Analog Converter (fixed frequency) ODE? E T t O N We need something more : an arbitrarily fast-growing ODE. 19 / 24

A less simplified proof binary stream generator : digits of α ∈ R 1 1 1 1 0 0 0 0 t 2 tanh( µ sin( 2 απ 4 round( t − 1 / 4 ,λ ) + 4 π/ 3 )) f ( α, µ, λ, t ) = 1 2 + 1 It’s horrible, but generable round is the mysterious rounding function... 20 / 24

A less simplified proof binary stream generator : digits of α ∈ R 1 1 1 1 0 0 0 0 t d 0 d 1 d 2 d 3 t a 0 a 1 a 2 a 3 dyadic stream generator : d i = m i 2 − d i , a i = 9 i + � j < i d j f ( α, γ, t ) = sin( 2 απ 2 round( t − 1 / 4 ,γ ) )) round is the mysterious rounding function... 20 / 24

A less simplified proof 1 1 1 1 0 0 0 0 t d 0 d 1 d 3 d 2 t a 0 a 1 a 2 a 3 20 / 24

A less simplified proof 1 1 1 1 0 0 0 0 t copy signal d 0 d 1 d 3 d 2 t a 0 a 1 a 2 a 3 20 / 24

A less simplified proof 1 1 1 1 0 0 0 0 t copy signal copy signal d 0 d 1 d 3 d 2 t a 0 a 1 a 2 a 3 20 / 24

A less simplified proof 1 1 1 1 0 0 0 0 t copy signal copy signal copy signal d 0 d 1 d 3 d 2 t a 0 a 1 a 2 a 3 20 / 24

A less simplified proof 1 1 1 1 0 0 0 0 t copy signal copy signal copy signal copy signal d 0 d 1 d 3 d 2 t a 0 a 1 a 2 a 3 20 / 24

A less simplified proof 1 1 1 1 0 0 0 0 t copy signal copy signal copy signal copy signal d 0 d 1 d 3 d 2 t a 0 a 1 a 2 a 3 This copy operation is the “non-trivial” part. 20 / 24

A less simplified proof t We can do almost piecewise constant functions... 20 / 24

A less simplified proof t We can do almost piecewise constant functions... ◮ ...that are bounded by 1... ◮ ...and have super slow changing frequency. 20 / 24

A less simplified proof t We can do almost piecewise constant functions... ◮ ...that are bounded by 1... ◮ ...and have super slow changing frequency. How do we go to arbitrarily large and growing functions? Can a polynomial ODE even have arbitrary growth? 20 / 24

An old question on growth Building a fast-growing ODE, that exists over R : y ′ 1 = y 1 y 1 ( t ) = exp( t ) � 21 / 24

An old question on growth Building a fast-growing ODE, that exists over R : y ′ 1 = y 1 y 1 ( t ) = exp( t ) � y ′ 2 = y 1 y 2 y 1 ( t ) = exp(exp( t )) � 21 / 24

An old question on growth Building a fast-growing ODE, that exists over R : y ′ 1 = y 1 y 1 ( t ) = exp( t ) � y ′ 2 = y 1 y 2 y 1 ( t ) = exp(exp( t )) � . . . . . . y ′ n = y 1 · · · y n y n ( t ) = exp( · · · exp( t ) · · · ) := e n ( t ) � 21 / 24

An old question on growth Building a fast-growing ODE, that exists over R : y ′ 1 = y 1 y 1 ( t ) = exp( t ) � y ′ 2 = y 1 y 2 y 1 ( t ) = exp(exp( t )) � . . . . . . y ′ n = y 1 · · · y n y n ( t ) = exp( · · · exp( t ) · · · ) := e n ( t ) � Conjecture (Emil Borel, 1899) With n variables, cannot do better than O t ( e n ( At k )) . 21 / 24

An old question on growth Counter-example (Vijayaraghavan, 1932) 1 2 − cos( t ) − cos( α t ) Sequence of arbitrarily growing spikes. t 21 / 24

An old question on growth Counter-example (Vijayaraghavan, 1932) 1 2 − cos( t ) − cos( α t ) Sequence of arbitrarily growing spikes. But not good enough for us. t 21 / 24

An old question on growth Theorem There exists a polynomial p : R d → R d such that for any continuous function f : R + → R , we can find α ∈ R d such that y ′ ( t ) = p ( y ( t )) y ( 0 ) = α, satisfies y 1 ( t ) � f ( t ) , ∀ t � 0 . 21 / 24

An old question on growth Theorem There exists a polynomial p : R d → R d such that for any continuous function f : R + → R , we can find α ∈ R d such that y ′ ( t ) = p ( y ( t )) y ( 0 ) = α, satisfies y 1 ( t ) � f ( t ) , ∀ t � 0 . Note : both results require α to be transcendental . Conjecture still open for rational (or algebraic) coefficients. 21 / 24

Proof gem : iteration with differential equations Assume f is generable, can we iterate f with an ODE? That is, build a generable y such that y ( x , n ) ≈ f [ n ] ( x ) for all n ∈ N 22 / 24

Proof gem : iteration with differential equations Assume f is generable, can we iterate f with an ODE? That is, build a generable y such that y ( x , n ) ≈ f [ n ] ( x ) for all n ∈ N f ( x ) x t 0 1 1 3 2 y ′ ≈ 0 2 2 z ′ ≈ f ( y ) − z 22 / 24

Proof gem : iteration with differential equations Assume f is generable, can we iterate f with an ODE? That is, build a generable y such that y ( x , n ) ≈ f [ n ] ( x ) for all n ∈ N f ( x ) x t 0 1 1 3 2 y ′ ≈ 0 y ′ ≈ z − y 2 2 z ′ ≈ f ( y ) − z z ′ ≈ 0 22 / 24

Proof gem : iteration with differential equations Assume f is generable, can we iterate f with an ODE? That is, build a generable y such that y ( x , n ) ≈ f [ n ] ( x ) for all n ∈ N f [ 2 ] ( x ) f ( x ) x t 0 1 1 3 2 y ′ ≈ 0 y ′ ≈ z − y 2 2 z ′ ≈ f ( y ) − z z ′ ≈ 0 22 / 24

Universal initial value problem (IVP) Notes : y 1 ( x ) ◮ system of ODEs, x ◮ y is analytic, ◮ we need d ≈ 300. Theorem There exists a fixed (vector of) polynomial p such that for any continuous functions f and ε , there exists α ∈ R d such that y ′ ( t ) = p ( y ( t )) y ( 0 ) = α, has a unique solution y : R → R d and ∀ t ∈ R , | y 1 ( t ) − f ( t ) | � ε ( t ) . Remark : α is usually transcendental, but computable from f and ε 23 / 24

Universal DAE revisited y 1 ( x ) x Theorem There exists a fixed polynomial p and k ∈ N such that for any continuous functions f and ε , there exists α 0 , . . . , α k ∈ R such that p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( 0 ) = α 0 , y ′ ( 0 ) = α 1 , . . . , y ( k ) ( 0 ) = α k has a unique analytic solution and this solution satisfies such that | y ( t ) − f ( t ) | � ε ( t ) . 24 / 24

Backup slides 25 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . y 1 ( x ) x Note : existence and unicity of y by Cauchy-Lipschitz theorem. 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f ( x ) = x ◮ identity y ′ = 1 y ( 0 ) = 0 , y ( x ) = x � 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f ( x ) = x 2 ◮ squaring y ′ y 1 ( x )= x 2 y 1 ( 0 )= 0 , 1 = 2 y 2 � y ′ y 2 ( 0 )= 0 , 2 = 1 y 2 ( x )= x � 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . ◮ n th power Example : f ( x ) = x n y ′ y 1 ( x )= x n y 1 ( 0 )= 0 , 1 = ny 2 � y ′ y 2 ( x )= x n − 1 y 2 ( 0 )= 0 , 2 = ( n − 1 ) y 3 � . . . . . . . . . y n ( 0 )= 0 , y n = 1 y n ( x )= x � 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f ( x ) = exp( x ) ◮ exponential y ′ = y y ( 0 )= 1 , y ( x )= exp( x ) � 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f ( x ) = sin( x ) or f ( x ) = cos( x ) ◮ sine/cosine y ′ y 1 ( 0 )= 0 , 1 = y 2 y 1 ( x )= sin( x ) � y ′ y 2 ( 0 )= 1 , 2 = − y 1 y 2 ( x )= cos( x ) � 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f ( x ) = tanh( x ) ◮ hyperbolic tangent y ′ = 1 − y 2 y ( 0 )= 0 , y ( x )= tanh( x ) � x tanh( x ) 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . 1 Example : f ( x ) = ◮ rational function 1 + x 2 f ′ ( x ) = ( 1 + x 2 ) 2 = − 2 xf ( x ) 2 − 2 x 1 y ′ 1 = − 2 y 2 y 2 y 1 ( 0 )= 1 , y 1 ( x )= � 1 1 + x 2 y ′ y 2 ( 0 )= 0 , 2 = 1 y 2 ( x )= x � 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f = g ± h ◮ sum/difference ( f ± g ) ′ = f ′ ± g ′ 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f = gh ◮ product ( gh ) ′ = g ′ h + gh ′ 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f = 1 ◮ inverse g f ′ = − g ′ g 2 = − g ′ f 2 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . � Example : f = g ◮ integral f ′ = g 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f = g ′ ◮ derivative f ′ = g ′′ = ( p 1 ( z )) ′ = ∇ p 1 ( z ) · z ′ 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f = g ◦ h ◮ composition ( z ◦ h ) ′ = ( z ′ ◦ h ) h ′ = p ( z ◦ h ) h ′ 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f ′ = tanh ◦ f ◮ Non-polynomial differential equation f ′′ = (tanh ′ ◦ f ) f ′ = ( 1 − (tanh ◦ f ) 2 ) f ′ 26 / 24

Generable functions (total, univariate) Types Definition f : R → R is generable if there exists ◮ d ∈ N : dimension d , p and y 0 such that the solution y to ◮ p ∈ R d [ R n ] : polynomial y ′ ( x ) = p ( y ( x )) y ( 0 ) = y 0 , vector ◮ y 0 ∈ R d , y : R → R d satisfies f ( x ) = y 1 ( x ) for all x ∈ R . Example : f ( 0 ) = f 0 , f ′ = g ◦ f ◮ Initial Value Problem (IVP) f ′ = g ′′ = ( p ( z )) ′ = ∇ p ( z ) · z ′ 26 / 24

Generable functions : a first summary Nice theory for the class of total and univariate generable functions : ◮ analytic ◮ contains polynomials, sin , cos , tanh , exp ◮ stable under ± , × , /, ◦ and Initial Value Problems (IVP) ◮ technicality on the field K of coefficients for stability under ◦ 27 / 24

Generable functions : a first summary Nice theory for the class of total and univariate generable functions : ◮ analytic ◮ contains polynomials, sin , cos , tanh , exp ◮ stable under ± , × , /, ◦ and Initial Value Problems (IVP) ◮ technicality on the field K of coefficients for stability under ◦ Limitations : ◮ total functions ◮ univariate 27 / 24

Generable functions (generalization) Types Definition f : X ⊆ R n → R is generable if X is open ◮ n ∈ N : input dimension ◮ d ∈ N : dimension connected and ∃ d , p , x 0 , y 0 , y such that ◮ p ∈ K d × d [ R d ] : y ( x 0 ) = y 0 , J y ( x ) = p ( y ( x )) polynomial matrix and f ( x ) = y 1 ( x ) for all x ∈ X . ◮ x 0 ∈ K n J y ( x ) = Jacobian matrix of y at x ◮ y 0 ∈ K d , y : X → R d Notes : ◮ Partial differential equation! ◮ Unicity of solution y ... ◮ ... but not existence (ie you have to show it exists) 28 / 24

Generable functions (generalization) Types Definition f : X ⊆ R n → R is generable if X is open ◮ n ∈ N : input dimension ◮ d ∈ N : dimension connected and ∃ d , p , x 0 , y 0 , y such that ◮ p ∈ K d × d [ R d ] : y ( x 0 ) = y 0 , J y ( x ) = p ( y ( x )) polynomial matrix and f ( x ) = y 1 ( x ) for all x ∈ X . ◮ x 0 ∈ K n J y ( x ) = Jacobian matrix of y at x ◮ y 0 ∈ K d , y : X → R d Example : f ( x 1 , x 2 ) = x 1 x 2 ( n = 2 , d = 3) ◮ monomial 2 y 2 x 1 x 2       0 3 y 2 y 3 3 2  , y ( 0 , 0 ) = 0 J y = 1 0 y ( x ) = x 1 �      0 0 1 x 2 28 / 24

Generable functions (generalization) Types Definition f : X ⊆ R n → R is generable if X is open ◮ n ∈ N : input dimension ◮ d ∈ N : dimension connected and ∃ d , p , x 0 , y 0 , y such that ◮ p ∈ K d × d [ R d ] : y ( x 0 ) = y 0 , J y ( x ) = p ( y ( x )) polynomial matrix and f ( x ) = y 1 ( x ) for all x ∈ X . ◮ x 0 ∈ K n J y ( x ) = Jacobian matrix of y at x ◮ y 0 ∈ K d , y : X → R d Example : f ( x 1 , x 2 ) = x 1 x 2 ◮ monomial 2 ∂ x 1 y 1 = y 2 y 1 ( x ) = x 1 x 2 y 1 ( 0 , 0 )= 0 , 3 , ∂ x 2 y 1 = 3 y 2 y 3 � 2 y 2 ( 0 , 0 )= 0 , ∂ x 1 y 2 = 1 , ∂ x 2 y 2 = 0 y 2 ( x ) = x 1 � y 3 ( 0 , 0 )= 0 , ∂ x 1 y 3 = 0 , ∂ x 2 y 3 = 1 y 3 ( x ) = x 2 � This is tedious! 28 / 24

Generable functions (generalization) Types Definition f : X ⊆ R n → R is generable if X is open ◮ n ∈ N : input dimension ◮ d ∈ N : dimension connected and ∃ d , p , x 0 , y 0 , y such that ◮ p ∈ K d × d [ R d ] : y ( x 0 ) = y 0 , J y ( x ) = p ( y ( x )) polynomial matrix and f ( x ) = y 1 ( x ) for all x ∈ X . ◮ x 0 ∈ K n J y ( x ) = Jacobian matrix of y at x ◮ y 0 ∈ K d , y : X → R d Last example : f ( x ) = 1 x for x ∈ ( 0 , ∞ ) ◮ inverse function ∂ x y = − y 2 y ( x ) = 1 y ( 1 )= 1 , � x 28 / 24

Generable functions : summary Nice theory for the class of multivariate generable functions (over connected domains) : ◮ analytic ◮ contains polynomials, sin , cos , tanh , exp ◮ stable under ± , × , /, ◦ and Initial Value Problems (IVP) ◮ technicality on the field K of coefficients for stability under ◦ 29 / 24