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Computing the Multiplicity Structure from Geometric Involutive Form Xiaoli Wu and Lihong Zhi Key Laboratory of Mathematics Mechanization Chinese Academy of Sciences Outline Compute an isolated primary component Bates etc.06, Corless


  1. Computing the Multiplicity Structure from Geometric Involutive Form Xiaoli Wu and Lihong Zhi Key Laboratory of Mathematics Mechanization Chinese Academy of Sciences

  2. Outline • Compute an isolated primary component Bates etc.’06, Corless etc.’98, Dayton’07, Moller and Stetter’00

  3. Outline • Compute an isolated primary component Bates etc.’06, Corless etc.’98, Dayton’07, Moller and Stetter’00 • Construct differential operators Damiano etc.’07, Dayton and Zeng’05, Marinari etc.’95,’96, Mourrain’96

  4. Outline • Compute an isolated primary component Bates etc.’06, Corless etc.’98, Dayton’07, Moller and Stetter’00 • Construct differential operators Damiano etc.’07, Dayton and Zeng’05, Marinari etc.’95,’96, Mourrain’96 • Refine an approximate singular solution Lercerf’02, Leykin etc.’05,’07, Ojika etc.’83,’87

  5. Notations Consider a polynomial system F ∈ C [ x ] = C [ x 1 ,..., x s ]  f 1 ( x 1 ,..., x s ) = 0 ,    f 2 ( x 1 ,..., x s ) = 0 ,   F : . . .     f t ( x 1 ,..., x s ) = 0 .  Let I = ( f 1 ,..., f t ) be the ideal generated by f 1 ,..., f t .

  6. • An ideal Q is primary if, for any f , g ∈ C [ x ] , ⇒ f ∈ Q or ∃ m ∈ N , g m ∈ Q fg ∈ Q =

  7. • An ideal Q is primary if, for any f , g ∈ C [ x ] , ⇒ f ∈ Q or ∃ m ∈ N , g m ∈ Q fg ∈ Q = • Every ideal has an irredundant primary decomposition I = ∩ r i = 1 Q i , and Q i � ∩ i � = j Q j Q j is called primary component (ideal) of I .

  8. • An ideal Q is primary if, for any f , g ∈ C [ x ] , ⇒ f ∈ Q or ∃ m ∈ N , g m ∈ Q fg ∈ Q = • Every ideal has an irredundant primary decomposition I = ∩ r i = 1 Q i , and Q i � ∩ i � = j Q j Q j is called primary component (ideal) of I . • The minimal nonnegative integer ρ s.t. √ Q ρ ⊂ Q is called the index of Q .

  9. Theorem 1. [Van Der Waerden 1970] Suppose the polynomial ideal I has an isolated primary component Q whose associated prime P is maximal, and ρ is the index of Q, µ is the multiplicity. • If σ < ρ , then dim ( C [ x ] / ( I , P σ − 1 )) < dim ( C [ x ] / ( I , P σ )) • If σ ≥ ρ , then Q = ( I , P ρ ) = ( I , P σ ) Corollary 2. The index is less than or equal to the multiplicity ρ ≤ µ = dim ( C [ x ] / Q )

  10. Coefficient Matrix F can be written in terms of its coefficient matrix M ( 0 ) as d     x d 0 1 x d − 1     x 2 0 1     . .     . . . .         x 2     0 M ( 0 ) s     · =     d x 1 0         . . . .     . .         x s 0         1 0

  11. Prolongation • Successive prolongations yield F ( 0 ) = F , F ( 1 ) = F ∪ x 1 F ∪···∪ x s F ,... M ( 0 ) · v d = 0 , M ( 1 ) · v d + 1 = 0 ,... d d � T . where v i = x i , x i − 1 ,..., x , 1 �

  12. Prolongation • Successive prolongations yield F ( 0 ) = F , F ( 1 ) = F ∪ x 1 F ∪···∪ x s F ,... M ( 0 ) · v d = 0 , M ( 1 ) · v d + 1 = 0 ,... d d � T . where v i = x i , x i − 1 ,..., x , 1 � • dim F ( 0 ) = dim Nullspace ( M ( 0 ) d )

  13. Geometric Projection • A single geometric projection is defined as � d · [ x d ,..., 1 ] T = 0 � [ x d − 1 ,..., 1 ] ∈ C N d − 1 |∃ x d , M ( 0 ) π ( F ) =

  14. Geometric Projection • A single geometric projection is defined as � d · [ x d ,..., 1 ] T = 0 � [ x d − 1 ,..., 1 ] ∈ C N d − 1 |∃ x d , M ( 0 ) π ( F ) = • dim π ( F ( 0 ) ) is the dimension of a linear space spanned by the null vectors of M ( 0 ) corresponding to the monomials of d the highest degree d being deleted.

  15. Criterion of Involution Theorem 2. [Zhi and Reid 2004] A zero dimensional polynomial system F is involutive at prolongation order m and projected order ℓ if and only if π ℓ ( F ( m ) ) satisfies the projected elimination test: dim π ℓ � F ( m ) � = dim π ℓ + 1 � F ( m + 1 ) � and the symbol involutive test: dim π ℓ � F ( m ) � = dim π ℓ + 1 � F ( m ) �

  16. SNEPSolver in [Zhi and Reid 2004] • For the tolerance τ , compute dim ˆ π ℓ ( F ( m ) ) by SVD.

  17. SNEPSolver in [Zhi and Reid 2004] • For the tolerance τ , compute dim ˆ π ℓ ( F ( m ) ) by SVD. π ℓ ( F ( m ) ) is • Seek the smallest m and largest ℓ such that ˆ approximately involutive.

  18. SNEPSolver in [Zhi and Reid 2004] • For the tolerance τ , compute dim ˆ π ℓ ( F ( m ) ) by SVD. π ℓ ( F ( m ) ) is • Seek the smallest m and largest ℓ such that ˆ approximately involutive. • The number of solutions of polynomial system F is d = dim ( C [ x ] / I ) = dim ˆ π ℓ ( F ( m ) ) .

  19. SNEPSolver in [Zhi and Reid 2004] • For the tolerance τ , compute dim ˆ π ℓ ( F ( m ) ) by SVD. π ℓ ( F ( m ) ) is • Seek the smallest m and largest ℓ such that ˆ approximately involutive. • The number of solutions of polynomial system F is d = dim ( C [ x ] / I ) = dim ˆ π ℓ ( F ( m ) ) . • The multiplication matrices M x 1 ,..., M x s are formed from π ℓ ( F ( m ) ) and ˆ π ℓ + 1 ( F ( m ) ) . the null vectors of ˆ

  20. Compute Primary Component I • Form the prime ideal P = ( x 1 − ˆ x 1 ,..., x s − ˆ x s ) .

  21. Compute Primary Component I • Form the prime ideal P = ( x 1 − ˆ x 1 ,..., x s − ˆ x s ) . • Compute d k = dim ( C [ x ] / ( I , P k )) by SNEPSolver for a given tolerance τ until d k = d k − 1 , set ρ = k − 1 , µ = d ρ , Q = ( I , P ρ ) .

  22. Compute Primary Component I • Form the prime ideal P = ( x 1 − ˆ x 1 ,..., x s − ˆ x s ) . • Compute d k = dim ( C [ x ] / ( I , P k )) by SNEPSolver for a given tolerance τ until d k = d k − 1 , set ρ = k − 1 , µ = d ρ , Q = ( I , P ρ ) . • Compute the multiplication matrices M x 1 ,..., M x s of C [ x ] / Q by SNEPSolver.

  23. Example 1 [Ojika 1987] I = ( f 1 = x 2 1 + x 2 − 3 , f 2 = x 1 + 0 . 125 x 2 2 − 1 . 5 ) ( 1 , 2 ) is a 3-fold solution. Form P = ( x 1 − 1 , x 2 − 2 ) .

  24. Example 1 [Ojika 1987] I = ( f 1 = x 2 1 + x 2 − 3 , f 2 = x 1 + 0 . 125 x 2 2 − 1 . 5 ) ( 1 , 2 ) is a 3-fold solution. Form P = ( x 1 − 1 , x 2 − 2 ) . • dim F ( 1 ) = dim F ( 2 ) ⇒ dim ( C [ x ] / ( I , P 2 )) = 2. = 2 = 2 2 • dim F ( 1 ) = dim F ( 2 ) ⇒ dim ( C [ x ] / ( I , P 3 )) = 3. = 3 = 3 3 • dim F ( 1 ) = dim F ( 2 ) ⇒ dim ( C [ x ] / ( I , P 4 )) = 3. = 3 = 4 4

  25. Example 1 [Ojika 1987] I = ( f 1 = x 2 1 + x 2 − 3 , f 2 = x 1 + 0 . 125 x 2 2 − 1 . 5 ) ( 1 , 2 ) is a 3-fold solution. Form P = ( x 1 − 1 , x 2 − 2 ) . • dim F ( 1 ) = dim F ( 2 ) ⇒ dim ( C [ x ] / ( I , P 2 )) = 2. = 2 = 2 2 • dim F ( 1 ) = dim F ( 2 ) ⇒ dim ( C [ x ] / ( I , P 3 )) = 3. = 3 = 3 3 • dim F ( 1 ) = dim F ( 2 ) ⇒ dim ( C [ x ] / ( I , P 4 )) = 3. = 3 = 4 4 Index ρ = 3, multiplicity µ = 3.

  26. Example 1 (continued) The multiplication matrices(local ring) w.r.t. { x 1 , x 2 , 1 } :     − 1 − 10 0 3 6 3 M x 1 =  , M x 2 = − 10 − 8 6 3 0 12        1 0 0 0 1 0

  27. Example 1 (continued) The multiplication matrices(local ring) w.r.t. { x 1 , x 2 , 1 } :     − 1 − 10 0 3 6 3 M x 1 =  , M x 2 = − 10 − 8 6 3 0 12        1 0 0 0 1 0 The primary component of I associating to ( 1 , 2 ) is { x 2 1 + x 2 − 3 , x 2 2 + 8 x 1 − 12 , x 1 x 2 − 6 x 1 − 3 x 2 + 10 }

  28. Differential Operators • Let D ( α ) = D ( α 1 ,..., α s ) : C [ x ] → C [ x ] denote the differential operator defined by: 1 α 1 ! ··· α s ! ∂ x α 1 1 ··· ∂ x α s D ( α 1 ,..., α s ) = s ,

  29. Differential Operators • Let D ( α ) = D ( α 1 ,..., α s ) : C [ x ] → C [ x ] denote the differential operator defined by: 1 α 1 ! ··· α s ! ∂ x α 1 1 ··· ∂ x α s D ( α 1 ,..., α s ) = s , • Let D = { D ( α ) , | α | ≥ 0 } , we define the space associated to x as I and ˆ △ ˆ x : = { L ∈ Span C ( D ) | L ( f ) | x = ˆ x = 0 , ∀ f ∈ I }

  30. Construct Differential Operators I • Write Taylor expansion of h ∈ C [ x ] at ˆ x : ∑ x 1 ) α 1 ··· ( x s − ˆ x s ) α s T ρ − 1 h ( x 1 ,..., x s ) = c α ( x 1 − ˆ α ∈ N s , | α | < ρ

  31. Construct Differential Operators I • Write Taylor expansion of h ∈ C [ x ] at ˆ x : ∑ x 1 ) α 1 ··· ( x s − ˆ x s ) α s T ρ − 1 h ( x 1 ,..., x s ) = c α ( x 1 − ˆ α ∈ N s , | α | < ρ x • Compute NF ( h ) , and expand it at ˆ NF ( h ( x )) = ∑ x ) β d β ( x − ˆ β and find scalars a αβ ∈ C such that d β = ∑ α a αβ c α .

  32. Construct Differential Operators I • Write Taylor expansion of h ∈ C [ x ] at ˆ x : ∑ x 1 ) α 1 ··· ( x s − ˆ x s ) α s T ρ − 1 h ( x 1 ,..., x s ) = c α ( x 1 − ˆ α ∈ N s , | α | < ρ x • Compute NF ( h ) , and expand it at ˆ NF ( h ( x )) = ∑ x ) β d β ( x − ˆ β and find scalars a αβ ∈ C such that d β = ∑ α a αβ c α . • For each β such that d β � = 0, return the operator L β = ∑ s = ∑ 1 α 1 ! ··· α s ! ∂ x α 1 1 ··· ∂ x α s a αβ D ( α ) . a αβ α α L = { L 1 ,..., L µ } is the set of differential operators.

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