Solving sparse polynomial systems using Gr¨ obner basis Mat´ ıas R. Bender Sorbonne Universit´ e, CNRS , INRIA , Laboratoire d’Informatique de Paris 6, LIP6 , ´ Equipe PolSys , 4 place Jussieu, F-75005, Paris, France Joint work with : Jean-Charles Faug` ere & Elias Tsigaridas
Resum´ e of the talk Objective Compute Gr¨ obner basis faster by exploiting the sparsity of the supports of the polynomials. We focus in the mixed case The polynomials have different supports. In this talk Algorithm to compute Gr¨ obner basis over semigroup algebras. Under regularity assumptions, no reductions to zero. Algorithm and complexity bounds to solve 0-dim. square systems. Improvements for special cases (mixed multihomogeneous & unmixed). Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 1 / 24
Gr¨ obner basics K [ x ] = K [ x 1 , . . . , x n ], polynomial ring in n indeterminates over K ⊂ C . Polynomial → � i c i x α ∈ K [ x ]. Monomial → x α , for α ∈ N n . Monomial ordering < Total order for monomials in K [ x ] such that, The monomial 1 is the smallest: ∀ x α � = 1, 1 < x α , Compatible with multiplication: for all x α , x β , x γ , x α < x β = ⇒ x α x γ < x β x γ Lexicographical (lex) y < x , 1 < y < y 2 < · · · < x < x y < x y 2 < · · · < x 2 < x 2 y < . . . . Degree lexicographical z < y < x , 1 < z < y < x < z 2 < y z < y 2 < x z < x y < x 2 < . . . . Degree reverse lexicographical order (grevlex) z < y < x , 1 < z < y < x < z 2 < y z < x z < y 2 < x y < x 2 < . . . Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 2 / 24
Gr¨ obner basics K [ x ] = K [ x 1 , . . . , x n ], polynomial ring in n indeterminates over K ⊂ C . Polynomial → � i c i x α ∈ K [ x ]. Monomial → x α , for α ∈ N n . Monomial ordering < Total order for monomials in K [ x ] such that, The monomial 1 is the smallest: ∀ x α � = 1, 1 < x α , Compatible with multiplication: for all x α , x β , x γ , x α < x β = ⇒ x α x γ < x β x γ Leading monomial → Biggest monomial (wrt > ) with non-zero coefficient. Gr¨ obner basis A subset G ⊂ I is a Gr¨ obner basis of the ideal I wrt > , if and only if, for every f ∈ I , there is g ∈ G such that LM > ( g ) divides LM > ( f ). Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 2 / 24
Computing Gr¨ obner basis : Lazard’s approach Compute Gr¨ obner basis for ( f 1 , f 2 , f 3 ) in K [ x , y ] wrt Grevlex( x > y ), f 1 := x + y + 1 f 2 := − x + y + 1 x 2 + x y − y 2 + x + y + 1 f 3 := Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 3 / 24
Computing Gr¨ obner basis : Lazard’s approach Compute Gr¨ obner basis for ( f 1 , f 2 , f 3 ) in K [ x , y ] wrt Grevlex( x > y ), f 1 := x + y + 1 f 2 := − x + y + 1 x 2 + x y − y 2 + x + y + 1 f 3 := We homogenize the system over K [ x , y , z ]. F 1 := x + y + z F 2 := − x + y + z x 2 + x y − y 2 + x z + y z + z 2 F 3 := Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 3 / 24
Computing Gr¨ obner basis : Lazard’s approach Compute Gr¨ obner basis for ( f 1 , f 2 , f 3 ) in K [ x , y ] wrt Grevlex( x > y ), f 1 := x + y + 1 f 2 := − x + y + 1 x 2 + x y − y 2 + x + y + 1 f 3 := We homogenize the system over K [ x , y , z ]. F 1 := x + y + z F 2 := − x + y + z x 2 + x y − y 2 + x z + y z + z 2 F 3 := For each d , compute triangular basis for � F 1 , F 2 , F 3 � d wrt Grevlex( x > y > z ). Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 3 / 24
Computing Gr¨ obner basis : Lazard’s approach Compute Gr¨ obner basis for ( f 1 , f 2 , f 3 ) in K [ x , y ] wrt Grevlex( x > y ), f 1 := x + y + 1 f 2 := − x + y + 1 x 2 + x y − y 2 + x + y + 1 f 3 := We homogenize the system over K [ x , y , z ]. F 1 := x + y + z F 2 := − x + y + z x 2 + x y − y 2 + x z + y z + z 2 F 3 := For each d , compute triangular basis for � F 1 , F 2 , F 3 � d wrt Grevlex( x > y > z ). Degree d = 1 , x y z F 1 1 1 1 F 2 − 1 1 1 Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 3 / 24
Computing Gr¨ obner basis : Lazard’s approach Compute Gr¨ obner basis for ( f 1 , f 2 , f 3 ) in K [ x , y ] wrt Grevlex( x > y ), f 1 := x + y + 1 f 2 := − x + y + 1 x 2 + x y − y 2 + x + y + 1 f 3 := We homogenize the system over K [ x , y , z ]. F 1 := x + y + z F 2 := − x + y + z x 2 + x y − y 2 + x z + y z + z 2 F 3 := For each d , compute triangular basis for � F 1 , F 2 , F 3 � d wrt Grevlex( x > y > z ). Degree d = 1 , x y z x y z − F 1 1 1 1 → F 1 1 1 1 F 2 − 1 1 1 F 2 + F 1 0 2 2 Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 3 / 24
Computing Gr¨ obner basis : Lazard’s approach for Grevlex For each degree d , compute triangular basis for � F 1 , . . . , F 3 � d : Degree d = 1 , x y z x y z − F 1 1 1 1 → F 1 1 1 1 F 2 − 1 1 1 F 2 + F 1 2 2 Degree d = 2, x 2 y 2 z 2 x y x z y z z F 1 1 1 1 1 1 1 y F 1 x F 1 1 1 1 − 1 1 1 z F 2 y F 2 − 1 1 1 − 1 1 1 x F 2 F 3 1 1 − 1 1 1 1 Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 4 / 24
Computing Gr¨ obner basis : Lazard’s approach for Grevlex For each degree d , compute triangular basis for � F 1 , . . . , F 3 � d : Degree d = 1 , x y z x y z − F 1 1 1 1 → F 1 1 1 1 F 2 − 1 1 1 F 2 + F 1 2 2 Degree d = 2, x 2 y 2 z 2 x y x z y z z F 1 1 1 1 1 1 1 y F 1 x F 1 1 1 1 z F 2 + z F 1 2 2 y F 2 + y F 1 2 2 ( x + y + z ) F 2 − ( x − y + z ) F 1 F 3 − ( x − y 2 + 1) F 1 + ( y 2 − 1) F 2 − 1 Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 4 / 24
Computing Gr¨ obner basis : Lazard’s approach for Grevlex For each degree d , compute triangular basis for � F 1 , . . . , F 3 � d : Degree d = 1 , x y z x y z − F 1 1 1 1 → F 1 1 1 1 F 2 − 1 1 1 F 2 + F 1 2 2 Degree d = 2, x 2 y 2 z 2 x y x z y z z F 1 1 1 1 1 1 1 y F 1 x F 1 1 1 1 z F 2 + z F 1 2 2 y F 2 + y F 1 2 2 ( x + y + z ) F 2 − ( x − y + z ) F 1 F 3 − ( x − y 2 + 1) F 1 + ( y 2 − 1) F 2 − 1 obner basis of � F 1 , F 2 , F 3 � → { x + y + z , y + z , z 2 } . Gr¨ Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 4 / 24
Computing Gr¨ obner basis : Lazard’s approach for Grevlex For each degree d , compute triangular basis for � F 1 , . . . , F 3 � d : Degree d = 1 , x y z x y z − F 1 1 1 1 → F 1 1 1 1 F 2 − 1 1 1 F 2 + F 1 2 2 Degree d = 2, x 2 y 2 z 2 x y x z y z z F 1 1 1 1 1 1 1 y F 1 x F 1 1 1 1 z F 2 + z F 1 2 2 y F 2 + y F 1 2 2 ( x + y + z ) F 2 − ( x − y + z ) F 1 F 3 − ( x − y 2 + 1) F 1 + ( y 2 − 1) F 2 − 1 obner basis of � F 1 , F 2 , F 3 � → { x + y + z , y + z , z 2 } . Gr¨ Its dehomogenization ( z = 1) is a Gr¨ obner basis of � f 1 , f 2 , f 3 � → { 1 } . Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 4 / 24
Complexity of Lazard’s algorithm Complexity depends on maximal degree. In generic coordinates, → Castelnuovo-Mumford (CM) regularity of I . Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 5 / 24
Complexity of Lazard’s algorithm Complexity depends on maximal degree. In generic coordinates, → Castelnuovo-Mumford (CM) regularity of I . Regular sequence ( F 1 , . . . , F m ) is a regular seq. ⇔ ∀ k ≤ m , F k is regular in K [ x ] / � F 1 , . . . , F k − 1 � . Macaulay bound If F 1 , . . . , F m regular sequence → CM regularity = � m i =1 deg ( f i ) − m + 1 Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 5 / 24
Complexity of Lazard’s algorithm Complexity depends on maximal degree. In generic coordinates, → Castelnuovo-Mumford (CM) regularity of I . Regular sequence ( F 1 , . . . , F m ) is a regular seq. ⇔ ∀ k ≤ m , F k is regular in K [ x ] / � F 1 , . . . , F k − 1 � . Macaulay bound If F 1 , . . . , F m regular sequence → CM regularity = � m i =1 deg ( f i ) − m + 1 Drawback: Many rows reduce to zero x 2 y 2 z 2 x y x z y z z F 1 1 1 1 y F 1 1 1 1 x F 1 1 1 1 z F 2 − 1 1 1 y F 2 + y F 1 2 2 ( x + y + z ) F 2 − ( x − y + z ) F 1 0 0 0 0 0 0 F 3 − ( x − y 2 + 1) F 1 + ( y 2 − 1) F 2 − 1 Mat´ ıas BENDER Gr¨ obner Basis & Sparse Systems April 2, 2019 5 / 24
Recommend
More recommend
