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 etc.’98, Dayton’07, Moller and Stetter’00
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
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
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 .
• An ideal Q is primary if, for any f , g ∈ C [ x ] , ⇒ f ∈ Q or ∃ m ∈ N , g m ∈ Q fg ∈ Q =
• 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 .
• 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 .
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 )
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
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 �
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 )
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 ) =
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.
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 ) �
SNEPSolver in [Zhi and Reid 2004] • For the tolerance τ , compute dim ˆ π ℓ ( F ( m ) ) by SVD.
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.
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 ) ) .
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
Compute Primary Component I • Form the prime ideal P = ( x 1 − ˆ x 1 ,..., x s − ˆ x s ) .
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 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.
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 ) .
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
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.
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
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 }
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 ,
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 }
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
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 .
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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