Additive Decompositions in Primitive Extensions Hao Du Key - PowerPoint PPT Presentation

Additive Decompositions in Primitive Extensions Hao Du Key Laboratory for Mathematics Mechanization AMSS, Chinese Academy of Sciences Joint work with Shaoshi Chen and Ziming Li ISSAC’18, July 16–19, New York, 2018

Outline Additive decomposition problem Previous results Additive decompositions in primitive extensions Hermite reduction Polynomial reduction Applications Du, Chinese Academy of Sciences Additive Decompositions 2/17

Terminologies Let F be a field of characteristic zero. A derivation on F is a map ′ : F → F s.t. for all a , b ∈ F , ( a + b ) ′ = a ′ + b ′ ( ab ) ′ = ab ′ + a ′ b . and ( F , ′ ) is a differential field. C F = { a ∈ F | a ′ = 0 } is the subfield of constants. A differential field ( E , D ) is a differential extension of F if F ⊆ E and D | F = ′ . Du, Chinese Academy of Sciences Additive Decompositions 3/17

Terminologies Let F be a field of characteristic zero. A derivation on F is a map ′ : F → F s.t. for all a , b ∈ F , ( a + b ) ′ = a ′ + b ′ ( ab ) ′ = ab ′ + a ′ b . and ( F , ′ ) is a differential field. C F = { a ∈ F | a ′ = 0 } is the subfield of constants. A differential field ( E , D ) is a differential extension of F if F ⊆ E and D | F = ′ . ′ = d / dx . Example. Set C ( x ) , C ( x , log( x )) , C ( x , e x ) , C ( x , √ x ) , . . . are differential fields. Du, Chinese Academy of Sciences Additive Decompositions 3/17

Additive decomposition problem Notation. F ′ := { f ′ | f ∈ F } . Problem. Given f ∈ F , find g , r ∈ F s.t. f = g ′ + r with the properties that f ∈ F ′ ⇐ ⇒ r = 0, r is minimal in some sense. Du, Chinese Academy of Sciences Additive Decompositions 4/17

Previous results Rational functions in C ( x ) (Ostrogradsky 1845, Hermite 1872) Rational functions in C ( x 1 , . . . , x n ) (Bostan, Lairez and Salvy 2013) Hyperexponential functions over C ( x ) (Bostan, Chen, Chyzak, Li and Xin 2013) Algebraic functions over C ( x ) (Chen, Kauers, Koutschan 2016) Fuchsian D-finite functions over C ( x ) (Chen, van Hoeij, Kauers, Koutschan 2017) D-finite functions over C ( x ) (van der Hoeven 2017, 2018, Bostan, Chyzak, Lairez and Salvy 2018) Du, Chinese Academy of Sciences Additive Decompositions 5/17

Primitive towers Definition. Let ( F , ′ ) ⊂ ( E , ′ ). t ∈ E is a primitive monomial if t ′ ∈ F , t is transcendental over F and C F ( t ) = C F . Examples. log( x ) and arctan( x ) are primitive monomials over C ( x ), � dx Li( x ):= log( x ) is a primitive monomial over C ( x , log( x )). Du, Chinese Academy of Sciences Additive Decompositions 6/17

Primitive towers Definition. Let ( F , ′ ) ⊂ ( E , ′ ). t ∈ E is a primitive monomial if t ′ ∈ F , t is transcendental over F and C F ( t ) = C F . Examples. log( x ) and arctan( x ) are primitive monomials over C ( x ), � dx Li( x ):= log( x ) is a primitive monomial over C ( x , log( x )). A primitive tower is F 0 ⊂ F 1 ⊂ · · · ⊂ F n � � � C ( x ) F 0 ( t 1 ) F n − 1 ( t n ) where t i is a primitive monomial over F i − 1 for all 1 ≤ i ≤ n . Du, Chinese Academy of Sciences Additive Decompositions 6/17

Hermite reduction Definition. Given a primitive tower F 0 ⊂ · · · ⊂ F n , p ∈ F n − 1 [ t n ] is t n -normal if gcd( p , p ′ ) ∈ F n − 1 ; f ∈ F n is t n -simple if f is proper and den( f ) is t n -normal. Du, Chinese Academy of Sciences Additive Decompositions 7/17

Hermite reduction Definition. Given a primitive tower F 0 ⊂ · · · ⊂ F n , p ∈ F n − 1 [ t n ] is t n -normal if gcd( p , p ′ ) ∈ F n − 1 ; f ∈ F n is t n -simple if f is proper and den( f ) is t n -normal. Lemma. For f ∈ F n , there exist g , h ∈ F n and p ∈ F n − 1 [ t n ] s.t. f = g ′ + h + p . where h is t n -simple. Moreover, f ∈ F ′ ⇒ = h = 0 . n Du, Chinese Academy of Sciences Additive Decompositions 7/17

Polynomial reduction Problem P. For p ∈ F n − 1 [ t n ], find g , q ∈ F n − 1 [ t n ] s.t. p = g ′ + q p ∈ F ′ n ⇐ ⇒ q = 0 . and Du, Chinese Academy of Sciences Additive Decompositions 8/17

Polynomial reduction Problem P. For p ∈ F n − 1 [ t n ], find g , q ∈ F n − 1 [ t n ] s.t. p = g ′ + q p ∈ F ′ n ⇐ ⇒ q = 0 . and Main idea. For a ∈ F n − 1 and d ∈ N , n = g ′ + q a t d with deg t n ( q ) < d . � a − c t ′ n ∈ F ′ for some c ∈ C . n − 1 Du, Chinese Academy of Sciences Additive Decompositions 8/17

Hermitian parts By Hermite reduction, for f ∈ F i , ∃ ! t i -simple h ∈ F i s.t. f = g ′ + h + p , where g ∈ F i and p ∈ F i − 1 [ t i ] for 1 ≤ i ≤ n . Definition. Call h the Hermitian part of f , denoted by hp t i ( f ). Du, Chinese Academy of Sciences Additive Decompositions 9/17

Hermitian parts By Hermite reduction, for f ∈ F i , ∃ ! t i -simple h ∈ F i s.t. f = g ′ + h + p , where g ∈ F i and p ∈ F i − 1 [ t i ] for 1 ≤ i ≤ n . Definition. Call h the Hermitian part of f , denoted by hp t i ( f ). If a − c t ′ n ∈ F ′ n − 1 and hp t n − 1 ( t ′ n ) � = 0, then hp t n − 1 ( a ) c = n ) . hp t n − 1 ( t ′ Du, Chinese Academy of Sciences Additive Decompositions 9/17

Straight towers Definition. A primitive tower F − 1 ⊂ F 0 ⊂ · · · ⊂ F n with F − 1 = C and F 0 = C ( t 0 ) is straight if hp t i − 1 ( t ′ i ) � = 0 for all 1 ≤ i ≤ n . Du, Chinese Academy of Sciences Additive Decompositions 10/17

Straight towers Definition. A primitive tower F − 1 ⊂ F 0 ⊂ · · · ⊂ F n with F − 1 = C and F 0 = C ( t 0 ) is straight if hp t i − 1 ( t ′ i ) � = 0 for all 1 ≤ i ≤ n . Define a t n -straight polynomial q ∈ F n − 1 [ t n ]: q is t 0 -straight if q = 0, q is t n -straight if lc t n ( q ) = u + v s.t. u ∈ F n − 1 is t n − 1 -simple, u � = c hp t n − 1 ( t ′ n ) for any nonzero c ∈ C , v ∈ F n − 2 [ t n − 1 ] is t n − 1 -straight. Du, Chinese Academy of Sciences Additive Decompositions 10/17

Straight towers Definition. A primitive tower F − 1 ⊂ F 0 ⊂ · · · ⊂ F n with F − 1 = C and F 0 = C ( t 0 ) is straight if hp t i − 1 ( t ′ i ) � = 0 for all 1 ≤ i ≤ n . Define a t n -straight polynomial q ∈ F n − 1 [ t n ]: q is t 0 -straight if q = 0, q is t n -straight if lc t n ( q ) = u + v s.t. u ∈ F n − 1 is t n − 1 -simple, u � = c hp t n − 1 ( t ′ n ) for any nonzero c ∈ C , v ∈ F n − 2 [ t n − 1 ] is t n − 1 -straight. Prop. Let q ∈ F n − 1 [ t n ] be t n -straight. Then q ∈ F ′ n ⇐ ⇒ q = 0. Du, Chinese Academy of Sciences Additive Decompositions 10/17

Flat towers Definition. A primitive tower ⊂ ⊂ · · · ⊂ F 0 F 1 F n � � � C ( x ) F 0 ( t 1 ) F n − 1 ( t n ) is flat if t ′ i ∈ F 0 for all 1 ≤ i ≤ n . Du, Chinese Academy of Sciences Additive Decompositions 11/17

Flat towers Definition. A primitive tower ⊂ ⊂ · · · ⊂ F 0 F 1 F n � � � C ( x ) F 0 ( t 1 ) F n − 1 ( t n ) is flat if t ′ i ∈ F 0 for all 1 ≤ i ≤ n . Notation. For 1 ≤ i ≤ n and p ∈ F i − 1 [ t i , . . . , t n ], hm i ( p ) is the head monomial of p w.r.t ≺ plex ( t i ≺ . . . ≺ t n ). hc i ( p ) is the head coefficient of p . Du, Chinese Academy of Sciences Additive Decompositions 11/17

Flat polynomials Definition. A polynomial q ∈ F n − 1 [ t n ] is t n -flat if: ∃ q i ∈ F i − 1 [ t i , . . . , t n ] s.t. q = � n i =1 q i , hc i ( q i ) is t i − 1 -simple for 1 ≤ i ≤ n , q 1 = 0 or hc 0 ( q 1 ) / ∈ span C { t ′ 1 , . . . , t ′ m } where  if hm 0 ( q 1 ) = 1 , n  m =  if hm 0 ( q 1 ) = t e s s · · · t e n n with e s > 0 s Du, Chinese Academy of Sciences Additive Decompositions 12/17

Flat polynomials Definition. A polynomial q ∈ F n − 1 [ t n ] is t n -flat if: ∃ q i ∈ F i − 1 [ t i , . . . , t n ] s.t. q = � n i =1 q i , hc i ( q i ) is t i − 1 -simple for 1 ≤ i ≤ n , q 1 = 0 or hc 0 ( q 1 ) / ∈ span C { t ′ 1 , . . . , t ′ m } where  if hm 0 ( q 1 ) = 1 , n  m =  if hm 0 ( q 1 ) = t e s s · · · t e n n with e s > 0 s Prop. Let q ∈ F n − 1 [ t n ] be t n -flat. Then q ∈ F ′ n ⇐ ⇒ q = 0. Du, Chinese Academy of Sciences Additive Decompositions 12/17

The main result Theorem. Given a straight (flat) tower F 0 ⊂ · · · ⊂ F n and f ∈ F n , there are g ∈ F n and q ∈ F n − 1 [ t n ] s.t. g ′ f = + hp t n ( f ) + q , ���� � �� � integrable non-integrable where q is t n -straight ( t n -flat). Moreover, f ∈ F ′ n ⇐ ⇒ hp t n ( f ) = q = 0, g ′ + ˜ q for t n -proper ˜ if f = ˜ h + ˜ h and ˜ q ∈ F n − 1 [ t n ], then  deg t n ( q ) ≤ deg t n (˜ q ) (straight)  den(hp t n ( f )) | den(˜ h ) and  q � plex ˜ q (flat) . Du, Chinese Academy of Sciences Additive Decompositions 13/17

Examples 1. Straight: � � 1 1 x f 1 = log( x )Li( x ) + log( x ) + Li( x ) − log( x ) ∈ C ( x , log( x ) , Li( x )) log( x ) 2. Flat: � arctan( x ) � 3 − log( x ) arctan( x ) 2 + log( x ) 2 ∈ C ( x , log( x ) , arctan( x )) f 2 = x 2 + 1 x Du, Chinese Academy of Sciences Additive Decompositions 14/17