On Continuity of the Roots of a Parametric Zero Dimensional - PowerPoint PPT Presentation

On Continuity of the Roots of a Parametric Zero Dimensional Multivariate Polynomial Ideal Yosuke Sato Ryoya Fukasaku Hiroshi Sekigawa Tokyo University of Science (Tokyo, Japan) 1 / 25

Motivation The roots of an unary polynomial are continuous in C m . 2 / 25

Motivation The roots of an unary polynomial are continuous in C m . Given parameters ¯ A “ A 1 ,..., A m and variables ¯ X “ X 1 ,..., X n , X s are continuous in C m ??? the roots of F Ă Q r ¯ A , ¯ 2 / 25

Motivation The roots of an unary polynomial are continuous in C m . Given parameters ¯ A “ A 1 ,..., A m and variables ¯ X “ X 1 ,..., X n , X s are continuous in C m ??? the roots of F Ă Q r ¯ A , ¯ Ex. Let V be the variety of t X 1 X 2 ` AX 2 ´ 1 , X 2 1 ` AX 2 ´ 1 u . tp ´ A ˘ ? , ´ A ˘ ? # 4 ` A 2 4 ` A 2 q , p 0 , 1 A qu p A � 0 q V “ 2 2 . tp˘ 1 , ˘ 1 qu p A “ 0 q - We can not treat p 0 , 1 A q in the case A “ 0 , and # 3 p A � 0 q - # p V q “ p A “ 0 q counting the multiplicities. □ 2 2 / 25

Motivation In this talk, each root is counted with multiplicity. To treat the continuity of the roots of F Ă Q r ¯ A , ¯ X s in S Ă C m , we have to assume that @ ¯ a P S # p V C p F p ¯ a qqq has the same cardinality, a , ¯ where V C p F p ¯ a qq is the variety of F p ¯ a q “ t f p ¯ X q : f P F u in C . 3 / 25

Motivation In this talk, each root is counted with multiplicity. To treat the continuity of the roots of F Ă Q r ¯ A , ¯ X s in S Ă C m , we have to assume that @ ¯ a P S # p V C p F p ¯ a qqq has the same cardinality, a , ¯ where V C p F p ¯ a qq is the variety of F p ¯ a q “ t f p ¯ X q : f P F u in C . For that, we use a Comprehensive Gr¨ obner System (CGS) which is an ideal tool for handling parametric ideals. 3 / 25

Motivation In this talk, each root is counted with multiplicity. To treat the continuity of the roots of F Ă Q r ¯ A , ¯ X s in S Ă C m , we have to assume that @ ¯ a P S # p V C p F p ¯ a qqq has the same cardinality, a , ¯ where V C p F p ¯ a qq is the variety of F p ¯ a q “ t f p ¯ X q : f P F u in C . For that, we use a Comprehensive Gr¨ obner System (CGS) which is an ideal tool for handling parametric ideals. By the result, we improve a quantifier elimination method. 3 / 25

Contents Motivation 1 Comprehensive Gr¨ obner System (CGS) 2 Continuity of Multivariate Roots 3 Quantifier Elimination (QE) using CGS; CGS-QE 4 Conclusion 5 4 / 25

Contents Motivation 1 Comprehensive Gr¨ obner System (CGS) 2 Continuity of Multivariate Roots 3 Quantifier Elimination (QE) using CGS; CGS-QE 4 Conclusion 5 5 / 25

Definition Def. 1. Let P Ă C m and S 1 ,..., S t Ă P and S “ t S 1 ,..., S t u . def ô the properties 1 and 2 are satisfied: S is a partition of P 1 Y t i “ 1 S i “ P and S i X S j “ H ( i � j ). 2 @ i D P , Q Ă Q r ¯ A s r S i “ V C p P qz V C p Q qs . □ 6 / 25

Definition Def. 1. Let P Ă C m and S 1 ,..., S t Ă P and S “ t S 1 ,..., S t u . def ô the properties 1 and 2 are satisfied: S is a partition of P 1 Y t i “ 1 S i “ P and S i X S j “ H ( i � j ). 2 @ i D P , Q Ă Q r ¯ A s r S i “ V C p P qz V C p Q qs . □ Fix a term order on the set of terms of ¯ X . HC g denotes the head coefficient of g P Q r ¯ A , ¯ X s . Rem. HC g P Q r ¯ A s for g P Q r ¯ A , ¯ X s . □ 6 / 25

Definition Def. 2. Let S 1 ,..., S t Ă P Ă C m and F , G 1 ,..., G t Ă Q r ¯ A , ¯ X s . def tp S 1 , G 1 q ,..., p S t , G t qu is a CGS of x F y on P ô for each ¯ a P S i 1 G i p ¯ a q is a Gr¨ obner Basis (GB) of x F p ¯ a qy , 2 @ g P G i p HC g p ¯ a q � 0 q , t S 1 ,..., S t u is a partition of P . □ Ex. Let f 1 “ AX 1 ` X 2 2 ´ 1 , f 2 “ X 3 2 ´ 1 . Then we obtain that tp V C p 0 qz V C p A q , t f 1 , f 2 uq , p V C p A q , t X 2 ´ 1 uqu is a CGS of x f 1 , f 2 y w.r.t. X 1 ą lex X 2 on C . □ 7 / 25

Well-known Fact dim p L q denotes the dimension of a linear space L . Rem. 3. Let G be a CGS. @p S , G q P G satisfies that for ¯ a P S - the set of leading terms of G p ¯ a q is invariant, so - dim p C r ¯ X s{x G p ¯ a qyq is invariant, so - dim p C r ¯ X s{x G p ¯ a qyq is finite ñ # p V C p G p ¯ a qqq is invariant, so - x G p ¯ a qy is zero-dimensional ñ # p V C p G p ¯ a qqq is invariant. We discuss the continuity of roots of G in S Ă C m such that x G p ¯ a qy is zero-dimensional for ¯ a P S . □ 8 / 25

Contents Motivation 1 Comprehensive Gr¨ obner System (CGS) 2 Continuity of Multivariate Roots 3 Quantifier Elimination (QE) using CGS; CGS-QE 4 Conclusion 5 9 / 25

Definition a “ p a 1 ,..., a n q P C n and ¯ b “ p b 1 ,..., b n q P C n , Def. 4. For ¯ b q def a , ¯ d p ¯ “ max p| a i ´ b i | : i q . □ 10 / 25

Definition a “ p a 1 ,..., a n q P C n and ¯ b “ p b 1 ,..., b n q P C n , Def. 4. For ¯ b q def a , ¯ d p ¯ “ max p| a i ´ b i | : i q . □ S l denotes the symmetric group of degree l . Def. 5. For α “ p α 1 ,...,α l q ,β “ p β 1 ,...,β l q P p C n q l , we define def α „ β ô D σ P S l @ i P t 1 ,..., l u r α i “ β σ p i q s , def def “t β Pp C n q l : α „ β u , p C n q M “t α M : α P C n u . an l -size multiset α M □ Ex. For a � b , p a , a , b q M “ p a , b , a q M , but p a , a , b q M � p a , b , b q M . □ 10 / 25

Definition Def. 6. For α M “ p α 1 ,...,α l q M , β M “ p β 1 ,...,β l q M P p C n q M , D p α M ,β M q def “ min p max p d p α i ,β σ p i q q : i q : σ P S l q . □ Ex. D pp 1 , 3 , 4 q M , p 2 , 3 , 5 q M q “ 1 . □ 11 / 25

Definition Def. 6. For α M “ p α 1 ,...,α l q M , β M “ p β 1 ,...,β l q M P p C n q M , D p α M ,β M q def “ min p max p d p α i ,β σ p i q q : i q : σ P S l q . □ Ex. D pp 1 , 3 , 4 q M , p 2 , 3 , 5 q M q “ 1 . □ Def. 7. Let S Ă C m , and I ◁ Q r ¯ A , ¯ X s be an ideal such that a P S r l “ dim p C r ¯ D l P N @ ¯ X s{ I p ¯ a qqs . Let θ p ¯ a q be the l -size multiset of the roots of I p ¯ a q for ¯ a P S . def ô @ ¯ a P S @ ϵ ą 0 D δ ą 0 Then, I has continuous roots on S @ ¯ a , ¯ a q ,θ p ¯ b P S r d p ¯ b q ă δ ñ D p θ p ¯ b qq ă ϵ s . □ 11 / 25

Main Theorem Th. 8. Let G be a CGS of I ◁ Q r ¯ A , ¯ X s , and p S , G q P G s.t. @ ¯ a P S x G p ¯ a qy is zero-dimensional. Then I has continuous roots on S . □ Ex. Let I “ x X 1 X 2 ` AX 2 ´ 1 , X 2 1 ` AX 2 ´ 1 y with a parameter A , S 1 “ V C p 0 qz V C p A q , S 2 “ V C p A q . I has continuous roots on S 1 and S 2 , since we get a CGS tp S 1 , t g 1 , g 2 , g 3 uq , p S 2 , t X 2 1 ´ 1 , X 2 ´ X 1 uqu , where g 1 “ X 3 1 ` AX 2 1 ` X 1 , g 2 “ AX 2 ` X 2 1 ´ 1 , g 3 “ X 1 X 2 ´ X 2 1 . □ 12 / 25

Main Theorem: Proof (Outline) a P S , we introduce l “ dim p C r ¯ X s{ I p ¯ a qq , Taking an arbitrary ¯ a qq s.t. α i “ p α p 1 q ,...,α p n q P V C p I p ¯ q , α i i i S Ñ p C n q M ; ¯ a ÞÑ p α 1 ,...,α l q M , θ : a ÞÑ p α p j q 1 ,...,α p j q : S Ñ C M ; ¯ l q M . π j 13 / 25

Main Theorem: Proof (Outline) a P S , we introduce l “ dim p C r ¯ X s{ I p ¯ a qq , Taking an arbitrary ¯ a qq s.t. α i “ p α p 1 q ,...,α p n q P V C p I p ¯ q , α i i i S Ñ p C n q M ; ¯ a ÞÑ p α 1 ,...,α l q M , θ : a ÞÑ p α p j q 1 ,...,α p j q : S Ñ C M ; ¯ l q M . π j 1 Each π j is continuous at ¯ a . ( ∵ G p ¯ a q is a GB of zero-dimensional I p ¯ a q ) 13 / 25

Main Theorem: Proof (Outline) a P S , we introduce l “ dim p C r ¯ X s{ I p ¯ a qq , Taking an arbitrary ¯ a qq s.t. α i “ p α p 1 q ,...,α p n q P V C p I p ¯ q , α i i i S Ñ p C n q M ; ¯ a ÞÑ p α 1 ,...,α l q M , θ : a ÞÑ p α p j q 1 ,...,α p j q : S Ñ C M ; ¯ l q M . π j 1 Each π j is continuous at ¯ a . ( ∵ G p ¯ a q is a GB of zero-dimensional I p ¯ a q ) 2 θ is continuous at ¯ a in the case α 1 “ ¨¨¨ “ α l . ( ∵ 1 ) 13 / 25

Main Theorem: Proof (Outline) a P S , we introduce l “ dim p C r ¯ X s{ I p ¯ a qq , Taking an arbitrary ¯ a qq s.t. α i “ p α p 1 q ,...,α p n q P V C p I p ¯ q , α i i i S Ñ p C n q M ; ¯ a ÞÑ p α 1 ,...,α l q M , θ : a ÞÑ p α p j q 1 ,...,α p j q : S Ñ C M ; ¯ l q M . π j 1 Each π j is continuous at ¯ a . ( ∵ G p ¯ a q is a GB of zero-dimensional I p ¯ a q ) 2 θ is continuous at ¯ a in the case α 1 “ ¨¨¨ “ α l . ( ∵ 1 ) a q“t α p 1 q 1 ,...,α p 1 q uˆ¨¨¨ˆt α p n q 1 ,...,α p n q We consider finite B p ¯ u . l l 13 / 25

Main Theorem: Proof (Outline): α i � α k X s s.t. @ α � α 1 P B p ¯ X q “ ř n a qr h p α q � h p α 1 qs , Let h p ¯ j “ 1 c j X j P Q r ¯ ϵ 0 ă min p| h p α q´ h p α 1 q| : α � α 1 P B p ¯ a qq , ¯ a , ¯ a q ,π j p ¯ b P S and δ ą 0 s.t. d p ¯ b q ă δ Ñ D p π j p ¯ b q ă ϵ 0 . 14 / 25

Main Theorem: Proof (Outline): α i � α k X s s.t. @ α � α 1 P B p ¯ X q “ ř n a qr h p α q � h p α 1 qs , Let h p ¯ j “ 1 c j X j P Q r ¯ ϵ 0 ă min p| h p α q´ h p α 1 q| : α � α 1 P B p ¯ a qq , ¯ a , ¯ a q ,π j p ¯ b P S and δ ą 0 s.t. d p ¯ b q ă δ Ñ D p π j p ¯ b q ă ϵ 0 . b qq D α 1 P B p ¯ 3 @ β i P V C p I p ¯ a q r d p α 1 ,β i q ă ϵ 0 s . ( ∵ 1 ) 14 / 25