Towards Mixed Gr obner Basis Algorithms: the Multihomogeneous and - PowerPoint PPT Presentation

Towards Mixed Gr¨ obner Basis Algorithms: the Multihomogeneous and Sparse Case July 19, 2018 Mat´ ıas R. Bender, Jean-Charles Faug` ere & Elias Tsigaridas Sorbonne Universit´ e, CNRS , INRIA , Laboratoire d’Informatique de Paris 6, LIP6 , ´ Equipe PolSys , 4 place Jussieu, F-75005, Paris, France

Computing Gr¨ obner basis over K [ x 1 , . . . , x n ] Consider ( f 1 , . . . , f r ) ∈ K [ x 1 , . . . , x n ]  Gr¨ obner basis Membership (Normal    Consider the ideal of f 1 , . . . , f r in  forms)  K [ x 1 , . . . , x n ]. Solving (Elimination)   There is a (finite) Gr¨ obner basis.   etc...  1/11

Computing Gr¨ obner basis over K [ x 1 , . . . , x n ] Consider ( f 1 , . . . , f r ) ∈ K [ x 1 , . . . , x n ]  Gr¨ obner basis Membership (Normal    Consider the ideal of f 1 , . . . , f r in  forms)  K [ x 1 , . . . , x n ]. Solving (Elimination)   There is a (finite) Gr¨ obner basis.   etc...  Computation If the homogenization of f 1 , . . . , f r (over K [ x 0 , x 1 , . . . , x n ]) is a regular sequence Avoid redundant computations (F5) Complexity bounds (Castelnuovo-Mumford Regularity) 1/11

Computing Gr¨ obner basis over K [ x 1 , . . . , x n ] Consider ( f 1 , . . . , f r ) ∈ K [ x 1 , . . . , x n ]  Gr¨ obner basis Membership (Normal    Consider the ideal of f 1 , . . . , f r in  forms)  K [ x 1 , . . . , x n ]. Solving (Elimination)   There is a (finite) Gr¨ obner basis.   etc...  Computation    If the homogenization of f 1 , . . . , f r     For sparse systems, the (over K [ x 0 , x 1 , . . . , x n ]) is a    regular sequence homogenization is NOT a regular sequence . Avoid redundant computations (F5)     Complexity bounds    (Castelnuovo-Mumford Regularity)   1/11

Sparse systems α c α X α − Support of f = � → Monomials in f , { α : c α � = 0 } . Sparse system − → The supports of the polynomials are “small”. 1 + xy + x 2 y + x 2 y 2 + x 3 y = 1 + 0 · x + 0 · y + 0 · x 2 + xy + 0 · y 2 + 0 · x 3 + x 2 y + 0 · xy 2 + 0 · y 3 + 0 · x 4 + x 3 y + x 2 y 2 + 0 · xy 3 + 0 · y 4 2/11

Sparse systems α c α X α − Support of f = � → Monomials in f , { α : c α � = 0 } . Sparse system − → The supports of the polynomials are “small”. Unmixed sparse system − → The polynomials have the same support. Mixed sparse system − → Different supports. 1 + xy + xy 2 + xy 3 1+ xy + x 2 y + x 2 y 2 + x 3 y 1+ x + xy + x 2 y + x 2 y 2 2/11

Previous work (Non-exhaustive!) Toric varieties [Demazure, 1970], [Hochster, 1971], [Satake, 1973], [Kempf, Knudsen, Mumford & Saint-Donat, 1973], [Miyake & Oda, 1975], [Ehlers, 1975], [Bernstein, 1975], [Kusnirenko, ∼ 1975] [Khovanskii, 1977], . . . . . . [Oda, 1988] . . . [Fulton, 1993] . . . [Cox, Little & Schenck, 2011] Sparse resultant [Gelfand, Kapranov & Zelevinsky, 1990], [Kapranov, Sturmfels & Zelevinsky, 1992], [Sturmfels, 1993], [Pedersen & Sturmfels, 1993], [Gelfand, Kapranov & Zelevinsky, 1994], [Canny & Emiris, 1995], [D´Andrea, 2002], [D´Andrea & Sombra, 2013] Sparse GB [Sturmfels, 1991], [Faug` ere, Spaenlehauer & Svartz, 2014] 3/11

Semigroup algebras: K [ S ] and K [ S h ] K [ S ] 4/11

Semigroup algebras: K [ S ] and K [ S h ] K [ S ] 4/11

Semigroup algebras: K [ S ] and K [ S h ] − → K [ S ] K [ S h ] 4/11

Semigroup algebras: K [ S ] and K [ S h ] − → K [ S ] K [ S h ] 4/11

Semigroup algebras: K [ S ] and K [ S h ] − → K [ S ] K [ S h ] 4/11

Semigroup algebras: K [ S ] and K [ S h ] − → K [ S ] K [ S h ] 4/11

Semigroup algebras: K [ S ] and K [ S h ] − → K [ S ] K [ S h ] 4/11

Semigroup algebras: K [ S ] and K [ S h ] − → K [ S ] K [ S h ] Compute GB over K [ S ] [Faug` ere, Spaenlehauer & Svartz, 2014] There is a Gr¨ obner basis. � If the homogenization of Generic unmixed systems → homogenization f 1 , . . . , f r is regular . f 1 , . . . , f r (over K [ S h ]) is a regular sequence No redundant Generic mixed systems → computations (F5) homogenization of f 1 , . . . , f r is Complexity bounds NOT a regular sequence . (C-M Regularity) 4/11

General approach for affine regular sequences For every k , given GB ( hom ( � f 1 , . . . , f k − 1 � )), compute GB ( J h k ) where J h k := hom ( � f 1 , . . . , f k − 1 � ) + � hom ( f k ) � . 5/11

General approach for affine regular sequences For every k , given GB ( hom ( � f 1 , . . . , f k − 1 � )), compute GB ( J h k ) where J h k := hom ( � f 1 , . . . , f k − 1 � ) + � hom ( f k ) � . � GB ( � f 1 ,..., f k � ) , and For GRevLex orders: GB ( J h k ) − → GB ( hom ( � f 1 , . . . , f k � )) . 5/11

General approach for affine regular sequences For every k , given GB ( hom ( � f 1 , . . . , f k − 1 � )), compute GB ( J h k ) where J h k := hom ( � f 1 , . . . , f k − 1 � ) + � hom ( f k ) � . � GB ( � f 1 ,..., f k � ) , and For GRevLex orders: GB ( J h k ) − → GB ( hom ( � f 1 , . . . , f k � )) . Lazard’s approach to GB ( J h k ), for each d ≤ reg ( J h k ), Compute a triangular basis of vector space ( J h k ) d . ( J h k ) d Gaussian elimination 5/11

General approach for affine regular sequences For every k , given GB ( hom ( � f 1 , . . . , f k − 1 � )), compute GB ( J h k ) where J h k := hom ( � f 1 , . . . , f k − 1 � ) + � hom ( f k ) � . � GB ( � f 1 ,..., f k � ) , and For GRevLex orders: GB ( J h k ) − → GB ( hom ( � f 1 , . . . , f k � )) . Lazard’s approach to GB ( J h k ), for each d , compute triangular basis of ( J h k ) d . ( J h k ) d Gaussian elimination 5/11

General approach for affine regular sequences For every k , given GB ( hom ( � f 1 , . . . , f k − 1 � )), compute GB ( J h k ) where J h k := hom ( � f 1 , . . . , f k − 1 � ) + � hom ( f k ) � . � GB ( � f 1 ,..., f k � ) , and For GRevLex orders: GB ( J h k ) − → GB ( hom ( � f 1 , . . . , f k � )) . Lazard’s approach to GB ( J h k ), for each d , compute triangular basis of ( J h k ) d . Affine F5 criterion , if ( f 1 , . . . , f r ) is an affine regular sequence , Reductions to zero in ( J h → polynomials in ( J h k ) d ← k − 1 ) d − deg ( f k ) . ( J h k ) d F5 Gauss. elim. 5/11

General approach for affine regular sequences For every k , given GB ( hom ( � f 1 , . . . , f k − 1 � )), compute GB ( J h k ) where J h k := hom ( � f 1 , . . . , f k − 1 � ) + � hom ( f k ) � . � GB ( � f 1 ,..., f k � ) , and For GRevLex orders: GB ( J h k ) − → GB ( hom ( � f 1 , . . . , f k � )) . Lazard’s approach to GB ( J h k ), for each d , compute triangular basis of ( J h k ) d . Affine F5 criterion, if ( f 1 , . . . , f r ) is an affine regular sequence , Reductions to zero in ( J h → polynomials in ( J h k ) d ← k − 1 ) d − deg ( f k ) . ( J h k ) d F5 Gauss. elim. We want to do the same over K [ S ] , we need GRevLex orders. 5/11

Sparse Degree sp ( X α ) = minimal s s.t. X α, s ∈ K [ S h ]. α c α X α , For f := � sp ( f ) = max ( { sp ( X α ) : c α � = 0 } ). Example sp ( x 2 y 2 + x 4 y 3 + x 6 y 5 ) = 4 sp ( x 2 y 2 ) = 2 ,   sp ( x 4 y 3 ) = 3 , sp ( x 6 y 5 ) = 4  6/11

Sparse Degree Sparse order sp ( X α ) = minimal s s.t. X α, s ∈ K [ S h ]. An order ≺ is compatible with the α c α X α , For f := � sparse degree ⇐ ⇒ ( ∀ α, β ∈ S ) if sp ( X α ) < sp ( X β ), then X α ≺ X β . sp ( f ) = max ( { sp ( X α ) : c α � = 0 } ). Example sp ( x 2 y 2 + x 4 y 3 + x 6 y 5 ) = 4 sp ( x 2 y 2 ) = 2 ,   sp ( x 4 y 3 ) = 3 , sp ( x 6 y 5 ) = 4  LM ≺ ( x 2 y 2 + x 4 y 3 + x 6 y 5 ) = x 6 y 5 6/11

Sparse orders Not “behave well” with multiplication. X α · LM ≺ ( f ) � = LM ≺ ( X α · f ) Not monomial orders. The division might not � LM ≺ ( y + x ) = y terminate. LM ≺ ( x · ( y + x )) = x 2 � = x · LM ≺ ( x + y ) 7/11

Sparse orders Not “behave well” with multiplication. X α · LM ≺ ( f ) � = LM ≺ ( X α · f ) Not monomial orders. The division might not � LM ≺ ( y + x ) = y terminate. LM ≺ ( x · ( y + x )) = x 2 � = x · LM ≺ ( x + y ) � X β X α ∈ K [ S ] X α divides X β , X α || X β , iff sp ( X α ) + sp ( X β X α ) = sp ( X β ) . 7/11

Sparse orders Not “behave well” with multiplication. X α · LM ≺ ( f ) � = LM ≺ ( X α · f ) Not monomial orders. The division might not � LM ≺ ( y + x ) = y terminate. LM ≺ ( x · ( y + x )) = x 2 � = x · LM ≺ ( x + y ) � X β X α ∈ K [ S ] X α divides X β , X α || X β , iff sp ( X α ) + sp ( X β X α ) = sp ( X β ) . 7/11

Sparse orders Not “behave well” with multiplication. X α · LM ≺ ( f ) � = LM ≺ ( X α · f ) Not monomial orders. The division might not � LM ≺ ( y + x ) = y terminate. LM ≺ ( x · ( y + x )) = x 2 � = x · LM ≺ ( x + y ) � X β X α ∈ K [ S ] X α divides X β , X α || X β , iff sp ( X α ) + sp ( X β X α ) = sp ( X β ) . The division algorithm terminates, X α � � If LM ≺ ( f ) || X α , then LM ≺ = X α . LM ≺ ( f ) · f 7/11