M ethodes formelles pour les equations aux d eriv ees - PowerPoint PPT Presentation

Multiple closed sets of monomials M := Mon( ∂ 1 , . . . , ∂ n ) := { ∂ i | i ∈ ( Z ≥ 0 ) n } S ⊆ M is if M -multiple closed m s ∈ S ∀ m ∈ M , s ∈ S M -multiple closed set ∂ 1 ∂ 2 ∂ 3 ∂ 4 generated by 2 , 1 ∂ 2 , 1 =: � ∂ 1 ∂ 2 2 , ∂ 3 1 ∂ 2 , ∂ 4 1 � M JNCF Luminy 2018

Multiple closed sets of monomials Lemma Every M -multiple closed set S ⊆ M has a finite generating set. Proof. F : m 1 , m 2 , m 3 . . . ∈ M m i � | m j ∀ i < j Every seq. s.t. is finite. Induction: n = 1 : clear. m 1 = ∂ a 1 1 · · · ∂ a n n − 1 → n : Let n . F ( j,d ) : m i = ∂ b 1 1 · · · ∂ d j · · · ∂ b n Define subsequence n � � { F ( j,d ) } = { F } We have: 1 ≤ j ≤ n 0 ≤ d ≤ a j { F ( j,d ) } are finite. By induction, the JNCF Luminy 2018

Multiple closed sets of monomials Lemma S ⊆ M Every M -multiple closed set has a finite generating set. Cor. Every ascending sequence of M -multiple closed sets becomes stationary. { p 1 , . . . , p r } Given a finite generating set for I � K [ ∂ 1 , . . . , ∂ n ] , Janet’s algorithm computes S 0 ⊆ S 1 ⊆ . . . ⊆ S k = lm( I ) (all M -multiple closed) where S 0 is generated by lm( p 1 ) , . . . , lm( p r ) ⇒ termination JNCF Luminy 2018

Decomposition into disjoint cones Def. Let C ⊆ Mon( ∂ 1 , . . . , ∂ n ) , µ ⊆ { ∂ 1 , . . . , ∂ n } . ∃ v ∈ C ( C, µ ) is a cone if s.t. C = Mon( µ ) v Variables in µ : multiplicative variables (for v ) Variables in { ∂ 1 , . . . , ∂ n } − µ : non-multiplicative variables (for v ) Dimension of ( C, µ ) : | µ | JNCF Luminy 2018

Decomposition into disjoint cones Def. Let S ⊆ M = Mon( ∂ 1 , . . . , ∂ n ) . { ( C 1 , µ 1 ) , . . . , ( C r , µ r ) } ⊂ P ( M ) × P ( { ∂ 1 , . . . , ∂ n } ) is a decomposition of S into disjoint cones if � r S = ˙ each ( C i , µ i ) is a cone and i =1 C i . { ( v 1 , µ 1 ) , . . . , ( v r , µ r ) } is a decomp. of S into disj. cones { (Mon( µ 1 ) v 1 , µ 1 ) , . . . , (Mon( µ r ) v r , µ r ) } if is one. JNCF Luminy 2018

Decomposition into disjoint cones Strategy of Janet’s algorithm: Decompose M -multiple closed sets S into disjoint cones. S = � ∂ 1 ∂ 2 2 , ∂ 3 1 ∂ 2 , ∂ 4 1 � M decomposition: ∂ 1 ∂ 2 { ∗ , ∂ 2 } 2 ∂ 2 1 ∂ 2 { ∗ , ∂ 2 } 2 ∂ 3 1 ∂ 2 { ∗ , ∂ 2 } ∂ 4 { ∂ 1 , ∂ 2 } 1 This can also be done for Mon( ∂ 1 , . . . , ∂ n ) − S . JNCF Luminy 2018

Janet division The possible ways of decomposing M -multiple closed sets into disjoint cones are studied as involutive divisions (Gerdt, Blinkov et. al.) Janet division: Let G ⊂ M = Mon( ∂ 1 , . . . , ∂ n ) be finite. v = ∂ a 1 1 · · · ∂ a n For a cone with vertex ∈ G n ∂ i is a iff multiplicative variable a i = max { b i | ∂ b ∈ G ; b j = a j ∀ j < i } . JNCF Luminy 2018

Janet division v = ∂ a 1 1 · · · ∂ a n For n : ⇒ a i = max { b i | ∂ b ∈ G ; b j = a j ∀ j < i } . ∂ i ∈ µ ⇐ G = { ∂ 2 ∂ 3 , ∂ 1 ∂ 2 ∂ 3 , ∂ 2 1 ∂ 2 ∂ 3 , ∂ 2 1 ∂ 2 Example: 2 } ∂ 2 ∂ 3 ∂ 1 ∂ 2 ∂ 3 ∂ 2 1 ∂ 2 ∂ 3 ∂ 2 1 ∂ 2 2 JNCF Luminy 2018

Janet division v = ∂ a 1 1 · · · ∂ a n For n : ⇒ a i = max { b i | ∂ b ∈ G ; b j = a j ∀ j < i } . ∂ i ∈ µ ⇐ G = { ∂ 2 ∂ 3 , ∂ 1 ∂ 2 ∂ 3 , ∂ 2 1 ∂ 2 ∂ 3 , ∂ 2 1 ∂ 2 Example: 2 } ∂ 2 ∂ 3 ∗ ∂ 2 ∂ 3 ∂ 1 ∂ 2 ∂ 3 ∗ ∂ 2 ∂ 3 ∂ 2 ∗ 1 ∂ 2 ∂ 3 ∂ 1 ∂ 3 ∂ 2 1 ∂ 2 ∂ 1 ∂ 2 ∂ 3 2 JNCF Luminy 2018

Decomposition into disjoint cones Decompose( G , η ) G ⊂ Mon( ∂ 1 , . . . , ∂ n ) , ∅ � = η ⊆ { ∂ 1 , . . . , ∂ n } G ← { g ∈ G |�∃ h ∈ G : h | g } if | G | ≤ 1 or | µ | = 1 then { ( m, η ) | m ∈ G } return else y ← y a with a = min { i | 1 ≤ i ≤ n, y i ∈ η } d ← max { deg y ( g ) | g ∈ G } G i ← { g ∈ G | deg y ( g ) = i } , i = 0 , . . . , d G i ← G i ∪ � i − 1 j =0 { y i − j g | g ∈ G j } , i = 1 , . . . , d T d ← { ( m, ζ ∪ { y } ) | ( m, ζ ) ∈ Decompose ( G d , η − { y } ) } T i ← Decompose ( G i , η − { y } ) , i = 0 , . . . , d − 1 � d return i =0 T i fi JNCF Luminy 2018

Janet reduction NF( p , T , > ) p ∈ K [ ∂ 1 , . . . , ∂ n ] , T = { ( d 1 , µ 1 ) , . . . , ( d s , µ s ) } r ← 0 while p � = 0 do if ∃ ( d, µ ) ∈ T : lm( p ) ∈ Mon( µ ) lm( d ) then p ← p − lc( p ) lm( p ) lm( d ) d lc( d ) else r ← r + lc( p ) lm( p ) p ← p − lc( p ) lm( p ) fi od return r Disjoint cones ⇒ course of algorithm is uniquely determined JNCF Luminy 2018

Janet’s algorithm JanetBasis( F , > ) F ⊆ K [ ∂ 1 , . . . , ∂ n ] finite G ← F do G ← auto-reduce G J ← { ( p 1 , µ 1 ) , . . . , ( p r , µ r ) } s.t. { (lm( p 1 ) , µ 1 ) , . . . , (lm( p r ) , µ r ) } Janet decomposition of � lm( G ) � M P ← { NF( ∂ · p, J ) | ( p, µ ) ∈ J, ∂ �∈ µ } (passivity check) G ← { p | ( p, µ ) ∈ J } ∪ P while P � = { 0 } return J JNCF Luminy 2018

Janet basis J = { ( p 1 , µ 1 ) , . . . , ( p r , µ r ) } for I = � F � Janet basis Invariant of the loop: G (or { p 1 , . . . , p r } ) always forms a gen. set for I r � Mon( µ i ) p i is a K -basis of I . i =1 Linear independence: clear. r � p ∈ I : p = c i p i i =1 JNCF Luminy 2018

Example g 1 := x 2 − y , Let I := � g 1 , g 2 � � K [ x, y ] , g 2 := xy − y . Let > be degrevlex, x > y . � lm( g 1 ) , lm( g 2 ) � : Decomposition into disjoint cones of { ( x 2 , { x, y } ) , ( xy, { y } ) } 2 � f := x · g 2 = x 2 y − xy ∈ I , f = c i g i ? i =1 g 3 := y 2 − y ∈ I Reduction of f modulo g 1 , g 2 yields: { ( g 1 , { x, y } ) , ( g 2 , { y } ) , ( g 3 , { y } ) } (minimal) Janet basis for I JNCF Luminy 2018

Linear PDEs Linearizing the system on nonlinear PDEs  ∂u  ∂x − u 2  = 0 ,  (4)  ∂ 2 u   ∂y 2 − u 3 = 0 , for one unknown function u of x and y , we obtain  ∂U   ∂x − 2 u U = 0 ,  (5) ∂ 2 U   ∂y 2 − 3 u 2 U  = 0 , for one unknown function U of x and y , where u is a solution of (4). Preparatory treatment of the nonlinear system (4) is necessary to deal with the linearized system (5). JNCF Luminy 2018

Linear PDEs Thomas decomposition � splitting system (4) into u x − u 2 = 0 { ∂ x , ∂ y } 2 u y 2 − u 4 = 0 { ∗ , ∂ y } u � = 0 { ∂ x , ∂ y } u = 0 √ Define D = Q ( 2)[ u, u x , u y , u x,x , u x,y , u y,y , . . . ] and the ideal I of D which consists of all D -linear combinations of � u x − u 2 � � u x − u 2 � � u x − u 2 � u x − u 2 , ∂ 2 ∂ x , ∂ y , , . . . x √ � √ 2 u 2 � � √ 2 u 2 � � √ 2 u 2 � 2 u 2 , ∂ x 2 2 2 2 , ∂ 2 u y − u y − , ∂ y u y − u y − , . . . x ⇒ D/I integral domain, define K = Quot( D/I ) , D = K � ∂ x , ∂ y � . √ √ 2 u 2 one may also choose u y + 2 2 2 u 2 .) (Instead of u y − JNCF Luminy 2018

Example � ∂ 2 U ∂ 3 U � � � ∂ 2 u ∂u ∂x U + u 2 ∂U ∂y 2 − 3 u 2 U = ∂x ∂y 2 − 3 ∂x ∂x ∂ 3 U ∂x ∂y 2 − 6 u 3 U − 6 u 3 U . = and ∂ 2 ∂ 3 U � ∂ 2 u ∂y + u ∂ 2 U � ∂U � � ∂y 2 U + 2 ∂u ∂U ∂x − 2 u U = ∂x ∂y 2 − 2 ∂y 2 ∂y 2 ∂y ∂ 3 U √ 2 u 2 ∂U ∂x ∂y 2 − 2 u 3 U − 2 ∂y − 6 u 3 U = Hence, we obtain � ∂ 2 U − ∂ 2 √ ∂ � � ∂U � 2 u 2 ∂U ∂y 2 − 3 u 2 U ∂y − 4 u 3 U , ∂x − 2 u U = 2 ∂y 2 ∂x which yields the consequence √ ∂U ∂y − 2 u U = 0 . JNCF Luminy 2018

Example Janet basis: U x − 2 u U = 0 , { ∂ x , ∂ y } , √ U y ∓ { ∗ , ∂ y } . 2 u U = 0 , Substituting 2 u ( x, y ) = √ 2 y + c, c ∈ R , − 2 x ± for u in this Janet basis results in a system of linear PDEs for U whose analytic solutions are given by C U ( x, y ) = √ 2 y + c ) 2 , C ∈ R . ( − 2 x ± JNCF Luminy 2018

(Generalized) Hilbert series J = { ( p 1 , µ 1 ) , . . . , ( p r , µ r ) } Janet basis for I We have lm( I ) = � lm( p 1 ) , . . . , lm( p r ) � M . Generalized Hilbert series � r � 1 H I ( ∂ 1 , . . . , ∂ n ) = lm( p i ) 1 − ∂ j i =1 ∂ j ∈ µ i enumerates a K -basis of � lm( I ) � . H I ( t, . . . , t ) is the usual Hilbert series. JNCF Luminy 2018

Example Janet basis for I J = { ( ∂ 1 ∂ 2 2 , { ∂ 2 } ) , ( ∂ 2 1 ∂ 2 2 , { ∂ 2 } ) , ( ∂ 3 1 ∂ 2 , { ∂ 2 } ) , ( ∂ 4 1 , { ∂ 1 , ∂ 2 } ) } generalized Hilbert series: ∂ 1 ∂ 2 + ∂ 2 1 ∂ 2 + ∂ 3 ∂ 4 1 ∂ 2 2 2 1 + 1 − ∂ 2 1 − ∂ 2 1 − ∂ 2 (1 − ∂ 1 )(1 − ∂ 2 ) JNCF Luminy 2018

Hilbert polynomial Janet basis of M : { ( p 1 , µ 1 ) , . . . , ( p r , µ r ) } � dim K M k t k H M ( t, . . . , t ) = k ≥ 0 � r 1 t deg( p i ) = (1 − t ) | µ i | i =1 � | µ i | + j − 1 � � r t deg( p i ) � t j = j i =1 j ≥ 0 Coeff. of t k in H M ( t, . . . , t ) ? For k ≥ max { deg( p i ) | i = 1 , . . . , r } : � | µ i | + k − deg( p i ) − 1 � r � dim K M k = k − deg( p i ) i =1 JNCF Luminy 2018

Example S = lm( I ) Mon( ∂ 1 , . . . , ∂ n ) − S Decomposition of into disjoint cones � generalized Hilbert series enum. a K -basis of K [ ∂ 1 , . . . , ∂ n ] /I generalized Hilbert series: 1 + ∂ 1 + ∂ 1 ∂ 2 + ∂ 2 1 + ∂ 2 1 ∂ 2 + ∂ 3 1 1 − ∂ 2 Hilbert polynomial: � | µ i | + k − deg( p i ) − 1 � 6 � = 1 k − deg( p i ) i =1 JNCF Luminy 2018

Power series solutions ∂ 2 u ∂x ∂y = 0 , { ∗ , ∂ y , ∂ z } , ∂ 3 u ∂x 2 ∂y = 0 , { ∗ , ∂ y , ∂ z } , ∂ 4 u ∂x 3 ∂z = 0 , { ∂ x , ∗ , ∂ z } , ∂ 4 u ∂x 3 ∂y = 0 , { ∂ x , ∂ y , ∂ z } . Janet decomposition of the set of parametric derivatives / generalized Hilbert series: 1 , { ∗ , ∂ y , ∂ z } , { ∗ , ∗ , ∂ z } , ∂ x , ∂ 2 ∂ 3 1 ∂ x (1 − ∂ y )(1 − ∂ z ) + 1 − ∂ z + 1 − ∂ z + x 1 − ∂ x . x ∂ 2 x , { ∗ , ∗ , ∂ z } , ∂ 3 x , { ∂ x , ∗ , ∗ } . Accordingly, a formal power series solution u is uniquely determined as u ( x, y, z ) = f 0 ( y, z ) + x f 1 ( z ) + x 2 f 2 ( z ) + x 3 f 3 ( x ) by any choice of formal power series f 0 , f 1 , f 2 , f 3 of the indicated variables. JNCF Luminy 2018

Power series solutions ∂ 2 u  = f ( x, y )   ∂x 2   ∂ 2 u  = g ( x, y )   ∂y 2  complete: ∂ 2 u  = f ( x, y ) { ∂ x , ∂ y } ,   ∂x 2      ∂ 2 u  = g ( x, y ) { ∗ , ∂ y } , ∂y 2    ∂ 3 u  ∂g   = { ∗ , ∂ y }   ∂x∂y 2 ∂x integrability condition: ∂ 2 � ∂ 2 u = ∂ 2 g �    ∂x 2 ∂y 2 ∂x 2 ∂y 2 = ∂ 2 g ∂ 2 f   = ⇒ ∂x 2 . ∂ 2 � ∂ 2 u = ∂ 2 f �    ∂y 2 ∂x 2 ∂y 2  JNCF Luminy 2018

Power series solutions Janet decomposition of the set of parametric derivatives: 1 , { ∗ , ∗ } , { ∗ , ∗ } , ∂ x , ∂ y , { ∗ , ∗ } , ∂ x ∂ y , { ∗ , ∗ } . initial conditions: u (0 , 0) = a 0 , 0 ,      ∂u   ∂x (0 , 0) = a 1 , 0 ,      ∂u ∂y (0 , 0) = a 0 , 1 ,       ∂ 2 u    ∂x∂y (0 , 0) = a 1 , 1   x i y j y j a 1 ,j x y j � � � u ( x, y ) = a 0 , 0 + a 0 , 1 y + a 1 , 0 x + a 1 , 1 x y + a i,j j ! + a 0 ,j j ! + i ! j ! i ≥ 2 , j ≥ 0 j ≥ 2 j ≥ 2 JNCF Luminy 2018

Power series solutions Janet decomposition of the set of parametric derivatives: 1 , { ∗ , ∗ } , ∂ x , { ∗ , ∗ } , ∂ y , { ∗ , ∗ } , { ∗ , ∗ } . ∂ x ∂ y , initial conditions: u (0 , 0) = a 0 , 0 ,      ∂u   ∂x (0 , 0) = a 1 , 0 ,      ∂u ∂y (0 , 0) = a 0 , 1 ,       ∂ 2 u    ∂x∂y (0 , 0) = a 1 , 1   � x � x � y � y � y � y ∂g u ( x, y ) = a 0 , 0 + ... + a 1 , 1 x y + f ( x, y ) d x d x + g (0 , y ) d y d y + x ∂x (0 , y ) d y d y 0 0 0 0 0 0 JNCF Luminy 2018

Desiderata Given a system of differential equations, we would like to be able to determine all analytic solutions; obtain an overview of all consequences of the system; in particular, given another differential equation, decide whether it is a consequence of the system or not; among the consequences find the ones which involve only certain specified unknowns. JNCF Luminy 2018

Elimination Lemma J ⊆ R := K [ X 1 , . . . , X n , Y 1 , . . . , Y m ] Janet basis w.r.t. any term order. For any 0 � = p ∈ R let lm( p ) be its leading monomial. If { p ∈ J | p ∈ K [ Y 1 , . . . , Y m ] } = { p ∈ J | lm( p ) ∈ K [ Y 1 , . . . , Y m ] } , J ∩ K [ Y 1 , . . . , Y m ] � J � ∩ K [ Y 1 , . . . , Y m ] . then generates Proof. Let 0 � = p ∈ � J � ∩ K [ Y 1 , . . . , Y m ] . Since J is a Janet basis, ∃ q ∈ J , lm( q ) ∈ K [ Y 1 , . . . , Y m ] , lm( q ) | lm( p ) . By assumption, q ∈ K [ Y 1 , . . . , Y m ] . p → 0 Reduction in K [ Y 1 , . . . , Y m ] . � JNCF Luminy 2018

Free resolution Janet basis J = { ( p 1 , µ 1 ) , . . . , ( p r , µ r ) } for I ∂ j p i = � We have k α i,j,k p k , ∂ j �∈ µ i , α i,j,k ∈ K [ µ k ] π : D | J | → D : ˆ Define p i �→ p i . ( ˆ p i std. gen.) Prop. � p i − ∂ j �∈ µ i , ∂ j ˆ α i,j,k ˆ p k , i = 1 , . . . , r, k form a Janet basis of ker π for a suitable monomial ordering. � construction of a free resolution of D/I . JNCF Luminy 2018

Example D = K [ ∂ 1 , ∂ 2 , ∂ 3 ] , ∂ 1 > ∂ 2 > ∂ 3 , I = ( ∂ 1 , ∂ 2 , ∂ 3 ) Janet basis: ∂ 1 ∂ 1 ∂ 2 ∂ 3 ∂ 2 ∗ ∂ 2 ∂ 3 ∂ 3 ∗ ∗ ∂ 3 ∂ 1 · ∂ 2 − ∂ 2 · ∂ 1 normal form computation: ∂ 1 · ∂ 3 − ∂ 3 · ∂ 1 ∂ 2 · ∂ 3 − ∂ 3 · ∂ 2     − ∂ 2 ∂ 1 0 ∂ 1 − ∂ 3 0 ∂ 1 ∂ 2         0 − ∂ 3 ∂ 2 ∂ 3 D 1 × 3 → D 1 × 3 − − − − − − − − − − − − − − − − − − − − − → D − → D/I − → 0 JNCF Luminy 2018

Example D = K [ ∂ 1 , ∂ 2 , ∂ 3 ] , ∂ 1 > ∂ 2 > ∂ 3 , I = ( ∂ 1 , ∂ 2 , ∂ 3 ) Janet basis: [ − ∂ 2 ∂ 1 0 ] ∂ 1 ∂ 2 ∂ 3 [ − ∂ 3 0 ∂ 1 ] ∂ 1 ∂ 2 ∂ 3 [ 0 − ∂ 3 ∂ 2 ] ∗ ∂ 2 ∂ 3 normal form computation: ∂ 1 · [ 0 − ∂ 3 ∂ 2 ] − ∂ 2 · [ − ∂ 3 0 ∂ 1 ] + ∂ 3 · [ − ∂ 2 ∂ 1 0]     − ∂ 2 ∂ 1 0 ∂ 1 − ∂ 3 0 ∂ 1 ∂ 2         0 → D ( ∂ 3 ∂ 1 ) 0 − ∂ 3 − ∂ 2 ∂ 2 ∂ 3 → D 1 × 3 → D 1 × 3 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − → D → D/I → 0 JNCF Luminy 2018

Free resolution Prop. � ∂ j ˆ p i − α i,j,k ˆ p k , ∂ j �∈ µ i , i = 1 , . . . , r, k form a Janet basis of ker π w.r.t. ≺ . ∂ 2 Choose a total order ≪ on J s.t. ∂ 1 ∂ 2 p k ≪ p l ∃ path from p l to p k in the Janet graph. ∂ 1 if ∂ 3 ∂ 1 � ∂ i lm( p k ) < ∂ j lm( p l ) ∂ i ˆ p k ≺ ∂ j ˆ p l : ⇐ ⇒ ∂ i lm( p k ) = ∂ j lm( p l ) or and p k ≪ p l JNCF Luminy 2018

Consequences { ( p 1 , µ 1 ) , . . . , ( p r , µ r ) } Janet basis for I � R can decide ideal membership normal form for residue classes modulo I enumeration of a K -basis of I and a K -basis of R/I (generalized Hilbert series) can easily determine Hilbert polynomial can read off a free resolution of R/I every Janet basis is a Gr¨ obner basis JNCF Luminy 2018

Janet bases over Z NF( p , T , ≺ ) p ∈ Z [ x 1 , . . . , x n ] , T = { ( d 1 , µ 1 ) , . . . , ( d l , µ l ) } r ← 0 while p � = 0 do if ∃ ( d, µ ) ∈ T : lm( p ) ∈ Mon( µ ) d then write lc( p ) = a · lc( d ) + b, | b | < | lc( d ) | a � = 0 if then p ← p − a lm( p ) lm( d ) d else move leading term from p to r fi else move leading term from p to r fi od return r JNCF Luminy 2018

Janet bases for Ore algebras Skew polynomial ring A [ ∂ ; σ, δ ] : A domain and K -algebra σ : A → A K -algebra endomorphism δ : A → A σ -derivation, i.e. δ ( a b ) = σ ( a ) δ ( b ) + δ ( a ) b, a, b ∈ A �� � a i ∂ i | a i ∈ A, i ∈ Z ≥ 0 A [ ∂ ; σ, δ ] = fin . with commutation rule a ∈ A ∂ a = σ ( a ) ∂ + δ ( a ) , JNCF Luminy 2018

Janet bases for Ore algebras Ore algebra D = A [ ∂ 1 ; σ 1 , δ 1 ] . . . [ ∂ m ; σ m , δ m ] : A = K or A = K [ x 1 , . . . , x n ] σ i : D → D K -algebra endomorphisms δ i : D → D σ i -derivations �� � a i ∂ i | a i ∈ A, i ∈ ( Z ≥ 0 ) m A [ ∂ 1 ; σ 1 , δ 1 ] . . . [ ∂ m ; σ m , δ m ] = fin . with commutation rules a ∈ A, ∂ i a = σ i ( a ) ∂ i + δ i ( a ) , ∂ i ∂ j = ∂ j ∂ i JNCF Luminy 2018

Janet bases for Ore algebras Weyl algebra: ordinary differential equations A 1 = K [ t ][ d dt a = a d d dt + da dt ] dt Weyl algebra: partial differential equations A n = K [ x 1 , . . . , x n ][ ∂ 1 , . . . , ∂ n ] ∂ i x j = x j ∂ i + δ ij B n = K ( x 1 , . . . , x n )[ ∂ 1 , . . . , ∂ n ] Shift operators: difference equations S h = K [ t ][ δ h ] δ h t = ( t − h ) δ h combinations . . . JNCF Luminy 2018

Janet bases for Ore algebras D = K [ x 1 , . . . , x n ][ ∂ 1 , . . . , ∂ m ] I left ideal of D generated by p 1 , . . . , p r normal form for elements of D : use ∂ i x j = σ i ( x j ) ∂ i + . . . to move all ∂ i to the right of every x j M := { x i ∂ j | i ∈ ( Z ≥ 0 ) n , j ∈ ( Z ≥ 0 ) m } consider M -multiple closed set generated by the normal forms of lm( p i ) , i = 1 , . . . , r decomp. into disj. cones as before reduction: all multiplications from the left JNCF Luminy 2018

Janet bases for Ore algebras D = K [ x 1 , . . . , x n ][ ∂ 1 , . . . , ∂ m ] I left ideal of D generated by p 1 , . . . , p r For termination of the algorithm, assume that ∂ i x j = ( c i,j x j + d i,j ) ∂ i + e i,j where c i,j ∈ K − { 0 } , d i,j ∈ K , e i,j ∈ K [ x 1 , . . . , x n ] deg( e i,j ) ≤ 1 with JNCF Luminy 2018

Involutive Janet (-like Gr¨ obner) bases for submodules of free modules over a commutative polynomial ring coefficients: rationals or finite fields and field extensions, and rational integers Janet division, Janet-like division term orderings: degrevlex, plex TOP / POT block / elimination orderings web: http://wwwb.math.rwth-aachen.de/Janet JNCF Luminy 2018

Involutive Analogues of Buchberger’s criteria can be selected Interface to C++: call fast routines when needed or switch to fast routines for the whole Maple session Syzygies, Hilbert series, etc. Applications: commutative algebra solving systems of algebraic equations web: http://wwwb.math.rwth-aachen.de/Janet JNCF Luminy 2018

Main procedures of Involutive InvolutiveBasis compute Janet(-like Gr¨ obner) basis PolInvReduce involutive reduction modulo Janet basis FactorModuleBasis vector space basis of residue class module Syzygies syzygy module PolResolution free resolution PolHilbertSeries , PolHilbertPolynomial , etc. combinatorial devices PolMinPoly , PolRepres , etc. computing in residue class rings JNCF Luminy 2018

Janet Janet (-like Gr¨ obner) bases for linear systems of partial differential equations Janet division, Janet-like division analogues of Buchberger’s criteria can be selected computational tools for differential operators elementary divisor algorithm for K ( x )[ ∂ ] (Jacobson normal form) parametric derivatives formal power series solutions, polynomial solutions web: http://wwwb.math.rwth-aachen.de/Janet JNCF Luminy 2018

Main procedures of Janet JanetBasis compute Janet(-like Gr¨ obner) basis InvReduce involutive reduction modulo Janet basis ParamDeriv parametric derivatives CompCond , Resolution compatibility conditions (syzygies) HilbertSeries , HilbertPolynomial , etc. combinatorial devices SolSeries , PolySol formal power series / polynomial solutions ElementaryDivisors Jacobson normal form JNCF Luminy 2018

ginv C++ module for Python comp. of Gr¨ obner bases using involutive algorithms polynomials, differential / difference equations open source software originated by V. P. Gerdt, Y. A. Blinkov contributions by LBfM coefficients: rationals or finite fields and some algebraic and transcendental field extensions term orderings: degrevlex (TOP / POT), lex, product orderings see web page for timings web: http://invo.jinr.ru JNCF Luminy 2018

ginv import ginv st = ginv.SystemType("Polynomial") im = ginv.MonomInterface("DegRevLex", st, [’x’, ’y’]) ic = ginv.CoeffInterface("GmpZ", st) ip = ginv.PolyInterface("PolyList", st, im, ic) iw = ginv.WrapInterface("CritPartially", ip) iD = ginv.DivisionInterface("Janet", iw) eqs = ["x^2+y^2", ...] basis = ginv.basisBuild("TQ", iD, eqs) JNCF Luminy 2018

Involutive Basis Algorithm (Gerdt) f ∈ F ≺ choose with the lowest lm( f ) w.r.t. G ← { f } ; Q ← F − G do h ← 0 while Q � = ∅ and h = 0 do p ∈ Q ≺ choose with the lowest lm( p ) w.r.t. Q ← Q − { p } ; h ← NF( p, G, ≺ ) if h � = 0 then { g ∈ G | lm( g ) = x i lm( h ) , | i | > 0 } for all do Q ← Q ∪ { g } ; G ← G − { g } G ← G ∪ { h } Q ← Q ∪ { x · g | g ∈ G, x non-mult. for g } while Q � = ∅ return G JNCF Luminy 2018

Janet-like Gr¨ obner Bases Idea: do not store all the prolongations � Janet-like division Note: In general, the minimal Gr¨ obner basis is still a proper subset of the Janet-like Gr¨ obner basis. V. P. Gerdt, Y. A. Blinkov, Janet-like Monomial Division . Janet-like Gr¨ obner Bases . CASC 2005, LNCS 3781, Springer, 2005 JNCF Luminy 2018

Formal Power Series Let D = K [ z 1 , ..., z n ][ ∂ 1 , ..., ∂ l ] s.t. Janet bases can be computed. Theorem F := hom K ( D, K ) is an injective left D -module. The pairing ( , ) : D × F → K : ( d, λ ) �→ λ ( d ) is non-degenerate: λ ∈ F is uniquely determined by λ ( d ) , d ∈ D λ ∈ F is uniquely determined by λ ( m ) , m ∈ Mon( D ) JNCF Luminy 2018

Formal Power Series F := hom K ( D, K ) Thm. is an injective left D -module. Proof. By Baer’s criterion, consider w.l.o.g. left ideal I of D ϕ : I → F . ϕ : D → F ? and Extension � System of D -linear equations d · λ = ϕ ( d ) , d ∈ I , λ ∈ F . r � J Janet basis for I ⇒ Mon( µ i ) p i is a K -basis of I i =1 m ∈ Mon( D ) − lm( I ) choose values ( m, λ ) = λ ( m ) for ⇒ values (lm( p ) , λ ) uniquely determined ⇒ solution λ ∈ F ϕ � defined by ϕ (1) := λ . � JNCF Luminy 2018

References M. Janet, ees partielles , Le¸ cons sur les syst` emes d’´ equations aux d´ eriv´ Gauthiers-Villars, Paris, 1929 C. M´ eray, D´ emonstration g´ en´ erale de l’existence des int´ egrales des ´ equations aux d´ eriv´ ees partielles , J. de math´ ematiques pures et appliqu´ ees, 3e s´ erie, tome VI, 1880 C. Riquier, Les syst` emes d’´ equations aux d´ eriv´ ees partielles , Gauthiers-Villars, Paris, 1910 J. F. Ritt, Differential Algebra , Dover, 1966 JNCF Luminy 2018

References W. Plesken, D. Robertz, Janet’s approach to presentations and resolutions for polynomials and linear pdes , Archiv der Mathematik, 84 (1), 2005, pp. 22–37 Y. A. Blinkov, C. F. Cid, V. P. Gerdt, W. Plesken, D. Robertz, The Maple Package ”Janet”: I. Polynomial Systems and II. Linear Partial Differential Equations , Proceedings of CASC 2003, pp. 31–40 resp. pp. 41–54 F.-O. Schreyer, Die Berechnung von Syzygien mit dem verallgemeinerten Weierstraßschen Divisionssatz und eine Anwendung auf analytische at , Cohen-Macaulay-Stellenalgebren minimaler Multiplizit¨ Diploma Thesis, Univ. Hamburg, Germany, 1980 JNCF Luminy 2018

References D. Robertz, Formal Computational Methods for Control Theory , PhD thesis, RWTH Aachen University, 2006, available at http://darwin.bth.rwth-aachen.de/opus/volltexte/2006/1586 D. Robertz, Janet bases and applications , in: M. Rosenkranz, D. Wang, Gr¨ obner Bases in Symbolic Analysis , Radon Series Comp. Appl. Math., de Gruyter, 2007 D. Robertz, Noether normalization guided by monomial cone decompositions , J. of Symbolic Computation, 44 (10), 2009, pp. 1359–1373 D. Robertz, Formal Algorithmic Elimination for PDEs , Habilitationsschrift, accepted by the Faculty of Mathematics, Computer Science and Natural Sciences, RWTH Aachen University, 2012 JNCF Luminy 2018

References W. Plesken, D. Robertz, Constructing Invariants for Finite Groups , Experimental Mathematics, 14 (2), 2005, pp. 175–188 W. Plesken, D. Robertz, Representations, commutative algebra, and Hurwitz groups , J. Algebra, 300 (2006), 2006, pp. 223–247 W. Plesken, D. Robertz, Elimination for coefficients of special characteristic polynomials , Experimental Mathematics 17 (4), 2008, pp. 499–510 W. Plesken, D. Robertz, Linear Differential Elimination for Analytic Functions , Mathematics in Computer Science, 4 (2–3), 2010, pp. 231–242 JNCF Luminy 2018

References V. P. Gerdt, Y. A. Blinkov, Involutive bases of polynomial ideals. Minimal involutive bases , Mathematics and Computers in Simulation, 45, 1998 Y. A. Blinkov, V. P. Gerdt, D. A. Yanovich, Construction of Janet Bases, I. Monomial Bases, II. Polynomial Bases , Proceedings of CASC 2001 V. P. Gerdt, Involutive Algorithms for Computing Gr¨ obner Bases , Proc. “Computational commutative and non-commutative algebraic geometry” (Chishinau, June 6-11, 2004), IOS Press, 2005 V. P. Gerdt, Y. A. Blinkov, V. V. Mozzhilkin, Gr¨ obner Bases and Generation of Difference Schemes for Partial Differential Equations , Symmetry, Integrability and Geometry: Methods and Applications, 2006 JNCF Luminy 2018

References V. P. Gerdt, D. Robertz, A Maple Package for Computing Gr¨ obner Bases for Linear Recurrence Relations , Nuclear Instruments and Methods in Physics Research A, 559 (1), 2006, pp. 215–219 V. P. Gerdt, D. Robertz, Consistency of Finite Difference Approximations for Linear PDE Systems and its Algorithmic Verification , in: S. M. Watt (ed.), Proceedings of ISSAC 2010, TU M¨ unchen, Germany, pp. 53–59 V. P. Gerdt, D. Robertz, Computation of Difference Gr¨ obner Bases , Computer Science Journal of Moldova, 20 (2), 2012, pp. 203–226 JNCF Luminy 2018

References F. Chyzak, B. Salvy, Non-commutative elimination in Ore algebras proves multivariate identities , J. Symbolic Computation, 26, 1998 V. Levandovskyy, Non-commutative Computer Algebra for polynomial algebras: Gr¨ obner bases, applications and implementation , PhD thesis, Univ. Kaiserslautern, Germany, 2005 V. P. Gerdt, D. A. Yanovich, Experimental Analysis of Involutive Criteria , “Algorithmic Algebra and Logic 2005”, April 3-6, 2005, Passau, Germany J. Apel, R. Hemmecke, Detecting unnecessary reductions in an involutive basis computation , J. Symbolic Computation, 40, 2005 JNCF Luminy 2018

References W. W. Adams, P. Loustaunau, An Introduction to Gr¨ obner Bases , AMS, 1994 T. Becker and V. Weispfenning, Gr¨ obner Bases. A Computational Approach to Commutative Algebra , Springer, 1993 D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms , Springer, 1992 D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry , Springer, 1995 JNCF Luminy 2018

References B. Malgrange, Syst` emes ` a coefficients constants , S´ eminaire Bourbaki 246:79–89, 1962–63. U. Oberst, Multidimensional constant linear systems , Acta Appl. Math. 20:1–175, 1990. J.-F. Pommaret and A. Quadrat, Algebraic analysis of linear multidimensional control systems , IMA Journal of Control and Information 16 (3):275–297, 1999. J.-F. Pommaret and A. Quadrat, A functorial approach to the behavior of multidimensional control systems , Applied Mathematics and Computer Science, 13:7–13, 2003. J.-F. Pommaret Partial Differential Control Theory Kluwer, 2001 JNCF Luminy 2018

References M. Barakat and D. Robertz, homalg: A meta-package for homologial algebra , Journal of Algebra and Its Applications 7 (3):299–317, 2008. F. Chyzak and A. Quadrat and D. Robertz, OreModules : A symbolic package for the study of multidimensional linear systems , in: Chiasson, J. and Loiseau, J.-J. (eds.), Applications of Time-Delay Systems , LNCIS 352, 233–264, Springer, 2007. T. Cluzeau and A. Quadrat, OreMorphisms : A homological algebra package for factoring and decomposing linear functional systems , in: Loiseau, J.-J., Michiels, W., Niculescu, S.-I., Sipahi, R. (eds.), Topics in Time-Delay Systems: Analysis, Algorithms and Control , LNCIS, Springer, 2008. JNCF Luminy 2018

Algebraic Geometry y � R → R 2 � � x 2 + y 2 − 1 = 0 x t 2 +1 , t 2 − 1 2 t t �→ t 2 +1 y = t 2 − 1 2 t Eliminate t in x = t 2 + 1 , . . . t 2 + 1 JNCF Luminy 2018

Special Solutions ∂v ∂t + v · ∇ v − ν ∆ v + 1 ρ ∇ p = 0 (Navier-Stokes) ∂ρ ∂t + ∇ · ( ρv ) = 0 cylindrical coordinates r , θ , z , ρ ≡ 1 (incompressible flow) Ansatz: v i ( r, θ, z ) = f i ( r ) g i ( θ ) h i ( z ) , i = 1 , 2 , 3 u ∈ { v 1 , v 2 , v 3 } , PDE: uu x,y − u x u y = 0 , ( x, y ) ∈ { ( r, θ ) , ( r, z ) , ( θ, z ) } one of the many simple systems of the Thomas decomposition: � − ( t + c 2 ) F 1 ( t ) ( θ + c 1 ) r � r v ( t, r, θ, z ) = − 2( t + c 2 ) , , 0 , t + c 2 r F 1 ( t ) − ( t + c 2 ) 2 F 1 ( t ) 2 p ( t, r, θ, z ) = ( t + c 2 ) ln( r ) ˙ + (ln( r ) + ( θ + c 1 ) 2 ) F 1 ( t ) 2 r 2 (( θ + c 1 ) 2 − 3 4 ) r 2 + F 2 ( t ) − 2 ν ln( r ) + . 2( t + c 2 ) 2 t + c 2 JNCF Luminy 2018

2. Thomas decomposition of differential systems JNCF Luminy 2018

Some references J. M. Thomas, Differential Systems , AMS Colloquium Publications, vol. XXI, 1937. V. P. Gerdt, On decomposition of algebraic PDE systems into simple subsystems , Acta Appl. Math., 101(1-3):39–51, 2008. T. B¨ achler, V. P. Gerdt, M. Lange-Hegermann, D. Robertz, Algorithmic Thomas Decomposition of Algebraic and Differential Systems , J. Symbolic Computation 47(10):1233–1266, 2012. D. Robertz, Formal Algorithmic Elimination for PDEs , Lecture Notes in Mathematics, Vol. 2121, Springer, 2014. JNCF Luminy 2018

Some references F. Boulier, D. Lazard, F. Ollivier, M. Petitot, Representation for the radical of a finitely generated differential ideal , ISSAC 1995, pp. 158–166. D. Wang, Decomposing polynomial systems into simple systems , J. Symbolic Computation 25(3):295–314, 1998. E. Hubert, Notes on triangular sets and triangulation-decomposition algorithms . in: LNCS, Vol. 2630, 2003, pp. 1–39 and 40–87. F. Lemaire, M. Moreno Maza, Y. Xie, The RegularChains library in Maple , SIGSAM Bulletin 39(3):96–97, 2005. D. Grigoriev, Complexity of quantifier elimination in the theory of ordinary differential equations , in: LNCS, vol. 378, 1989, pp. 11–25. JNCF Luminy 2018

Systems of PDEs A differential system S is given by p 1 = 0 , p 2 = 0 , . . . , p s = 0 , q 1 � = 0 , q 2 � = 0 , . . . , q t � = 0 , where p 1 , ..., p s and q 1 , ..., q t are polynomials in u 1 , ..., u m of z 1 , ..., z n and their partial derivatives. Ω open and connected subset of C n with coordinates z 1 , . . . , z n The solution set of S on Ω is Sol Ω ( S ) := { f = ( f 1 , . . . , f m ) | f k : Ω → C analytic , k = 1 , . . . , m, p i ( f ) = 0 , q j ( f ) � = 0 , i = 1 , . . . , s, j = 1 , . . . , t } . Appropriate choice of Ω is possible only after formal treatment. JNCF Luminy 2018

Systems of PDEs A differential system S is given by p 1 = 0 , p 2 = 0 , . . . , p s = 0 , q 1 � = 0 , q 2 � = 0 , . . . , q t � = 0 , Consequences of the system obtained in a finite number of steps from: p 1 = 0 , p 2 = 0 , . . . , p s = 0 are consequences, if p = 0 is consequence, then any partial derivative of p = 0 is, if p · q = 0 is consequence and q a factor of some q i , then p = 0 is consequence, if p = 0 , r = 0 are consequences, then a p + b r = 0 is ( a , b differential polynomials) JNCF Luminy 2018

Polynomial ODEs / PDEs � du � 2 − 4 t du dt − 4 u + 8 t 2 = 0 find: u = u ( t ) analytic dt u ( t ) = a 0 + a 1 t + a 2 t 2 2! + a 3 t 3 3! + . . . Substitute and compare coefficients:  a 2 1 − 4 a 0 = 0 a 0 := 0 ⇒ a 1 = 0      2 a 1 a 2 − 8 a 1 = 0 a 1 a 3 + a 2 2 − 6 a 2 + 8 = 0 ⇒ ( a 2 − 2)( a 2 − 4) = 0     .  . . Many case distinctions? Thomas’ algorithm � finitely many so-called simple systems (Joseph Miller Thomas, ∼ 1930) JNCF Luminy 2018

Algebraic geometry L = { p 1 ( x 1 , ..., x n ) = 0 , ..., p r = 0 , q 1 � = 0 , ..., q s � = 0 } polynomial equations (and inequations) Sol( L ) = { a ∈ C n | p i ( a ) = 0 , q j ( a ) � = 0 ∀ i, j } Conversely, let S ⊆ C n . I ( S ) = { p ∈ C [ x 1 , ..., x n ] | p ( a ) = 0 ∀ a ∈ S } Nullstellensatz (Hilbert, 1893) (for equations) zero sets in C n radical ideals of C [ x 1 , ..., x n ] ← → are bijections which are inverse to each other. JNCF Luminy 2018

Differential algebraic geometry Differential algebra (Ritt, Kolchin, Seidenberg, . . . ) Q ⊆ K a differential field with commuting derivations ∂ 1 , ..., ∂ n Differential polynomial ring with derivations ∂ 1 , ..., ∂ n K { u } := K [ ∂ i 1 1 · · · ∂ i n n u | i ∈ ( Z ≥ 0 ) n ] = K [ u, u z 1 , ..., u z n , u z 1 ,z 1 , ... ] K { u } not Noetherian (e.g., [ u ′ u ′′ , u ′′ u ′′′ , . . . ] ⊆ K { u } not fin. gen.) Thm. (Ritt-Raudenbush). Every radical diff. ideal of K { u 1 , . . . , u m } is finitely generated, is intersection of finitely many prime diff. ideals. Thm. (Differential Nullstellensatz). Every radical diff. ideal I � K { u 1 , . . . , u m } has a zero in a diff. field ext. If f ∈ K { u 1 , . . . , u m } vanishes for all zeros of I , then f ∈ I . of K . JNCF Luminy 2018

Thomas Decomposition K { u } = K [ u, u x , u y , . . . , u x,x , u x,y , u y,y , . . . ] diff. polynomial ring u < . . . < u y < u x < . . . < u y,y < u x,y < u x,x < . . . (ranking) u 3 algebraic reduction: p = x,x,y + . . . q = c u 2 x,x,y + . . . p → r = c · p − u x,x,y · q u 3 differential reduction: p = x,x,y,y + . . . q = c u 2 x,x,y + . . . ∂q ∂ y q = ∂u x,x,y u x,x,y,y + . . . ∂u x,x,y · p − u 2 ∂q p → r = x,x,y,y · ∂ y q ∂q reduction requires: initial c � = 0 and separant ∂u x,x,y � = 0 JNCF Luminy 2018

Thomas Decomposition R = K { u 1 , . . . , u m } Def. Thomas decomposition of diff. system S (or Sol ( S ) ): Sol ( S ) = Sol ( S 1 ) ⊎ . . . ⊎ Sol ( S r ) , S i simple diff. system Thm. S = { p 1 = 0 , ..., p s = 0 , q 1 � = 0 , ..., q t � = 0 } simple diff. system E diff. ideal generated by p 1 , . . . , p s q product of initials and separants of all p i Then E : q ∞ := { p ∈ R | q r · p ∈ E for some r ∈ Z ≥ 0 } = I R ( Sol ( S )) consists of all diff. polynomials in R vanishing on Sol ( S ) . JNCF Luminy 2018

Thomas Decomposition p = x 3 + (3 y + 1) x 2 + (3 y 2 + 2 y ) x + y 3 = 0 y x JNCF Luminy 2018

Thomas Decomposition p = x 3 + (3 y + 1) x 2 + (3 y 2 + 2 y ) x + y 3 = 0 y x JNCF Luminy 2018

Thomas Decomposition p = x 3 + (3 y + 1) x 2 + (3 y 2 + 2 y ) x + y 3 = 0 y x disc x ( p ) = y 2 (4 − 27 y 2 ) JNCF Luminy 2018

Thomas Decomposition p = x 3 + (3 y + 1) x 2 + (3 y 2 + 2 y ) x + y 3 = 0 y 2 non-real points x disc x ( p ) = y 2 (4 − 27 y 2 ) JNCF Luminy 2018

Simple Systems K field of char. 0 , p 1 , . . . , p s , q 1 , . . . , q t ∈ K [ x 1 , . . . , x n ] � � n � � p i ( a ) = 0 , q j ( a ) � = 0 V = a ∈ K ∀ i, j n − ( i − 1) − n − i : ( a i , a i +1 , . . . , a n ) �− π i : K → K → ( a i +1 , . . . , a n ) V 1 := V , V i +1 := π i ( V i ) V is simple , if for each i one of the following three cases holds:  ∃ ! a (1) i , . . . , a ( e ) ( a ( j )  ∃ e ∀ ( a i +1 , . . . , a n ) ∈ π i ( V i ) i , a i +1 , . . . , a n ) ∈ V i ,   i   ∃ ! a (1) i , . . . , a ( f ) ( a ( j ) ∃ f ∀ ( a i +1 , . . . , a n ) ∈ π i ( V i ) i , a i +1 , . . . , a n ) �∈ V i ,  i     ∀ ( a i +1 , . . . , a n ) ∈ π i ( V i ) ( a i , a i +1 , . . . , a n ) ∈ V i ∀ a i ∈ K Write V = W 1 ⊎ . . . ⊎ W r Thomas decomposition: where W j simple JNCF Luminy 2018

Simple Systems p 1 , . . . , p s , q 1 , . . . , q t ∈ K [ x 1 , . . . , x n ] , x 1 > x 2 > . . . > x n � � n � � p i ( a ) = 0 , q j ( a ) � = 0 V = a ∈ K ∀ i, j Identify K [ x 1 , . . . , x n ] = K [ x n ][ x n − 1 ] . . . [ x 1 ] . S = { p 1 = 0 , . . . , p s = 0 , q 1 � = 0 , . . . , q t � = 0 } is a simple system , if 1. Each variable is leader of at most one p i or q j . 2. The initial of p i , q j has no zero in π k ( V k ) , if x k is the leader of p i resp. q j . 3. p i ( x k , a k +1 , . . . , a n ) , q j ( x k , a k +1 , . . . , a n ) are square-free for all ( a k +1 , . . . , a n ) ∈ π k ( V k ) , if x k is the leader of p i resp. q j . JNCF Luminy 2018

Thomas Decomposition p = ax 2 + bx + c = 0 , p ∈ Q [ x, c, b, a ] , x > c > b > a ax 2 + bx + c = 0 JNCF Luminy 2018

Thomas Decomposition p = ax 2 + bx + c = 0 , p ∈ Q [ x, c, b, a ] , x > c > b > a ax 2 + bx + c = 0 bx + c = 0 a � = 0 a = 0 JNCF Luminy 2018

Thomas Decomposition p = ax 2 + bx + c = 0 , p ∈ Q [ x, c, b, a ] , x > c > b > a ax 2 + bx + c = 0 2 ax + b = 0 bx + c = 0 4 ac − b 2 � = 0 4 ac − b 2 = 0 a � = 0 a � = 0 a = 0 JNCF Luminy 2018