Signature-based algorithms for computing Gröbner bases over Principal Ideal Domains Maria Francis 1 , Thibaut Verron 2 1. Indian Institute of Technology Hyderabad, Hyderabad, India 2. Institute for Algebra, Johannes Kepler University, Linz, Austria Séminaire MAX, Laboratoire d’Informatique de l’École Polytechnique 18 février 2019 1
Gröbner bases ◮ Valuable tool for many questions related to polynomial equations (solving, elimination, dimension of the solutions...) ◮ Classically used for polynomials over fields ◮ Some applications with coefficients in general rings (elimination, combinatorics...) 2
Gröbner bases ◮ Valuable tool for many questions related to polynomial equations (solving, elimination, dimension of the solutions...) ◮ Classically used for polynomials over fields ◮ Some applications with coefficients in general rings (elimination, combinatorics...) Many algorithms for fields ◮ First algorithm: Buchberger (1965) ◮ Optimizations related to selection strategies: “Normal” (1985), “Sugar” (1991) ◮ Criteria: Buchberger’s coprime and chain criteria (1979), Gebauer-Möller (1988) ◮ Replace polynomial arithmetic with linear algebra: Lazard (1983), F4 (1999) ◮ Signature-based criteria: F5 (2002), GVW (2010)... 2
Gröbner bases ◮ Valuable tool for many questions related to polynomial equations (solving, elimination, dimension of the solutions...) ◮ Classically used for polynomials over fields ◮ Some applications with coefficients in general rings (elimination, combinatorics...) Many algorithms for fields ◮ First algorithm: Buchberger (1965) ◮ Optimizations related to selection strategies: “Normal” (1985), “Sugar” (1991) ◮ Criteria: Buchberger’s coprime and chain criteria (1979), Gebauer-Möller (1988) ◮ Replace polynomial arithmetic with linear algebra: Lazard (1983), F4 (1999) ◮ Signature-based criteria: F5 (2002), GVW (2010)... And for rings: ◮ Möller (1988) for general rings and principal domains, Kandri-Rodi Kapur (1988) for Euclidean domains... ◮ Optimizations and general criteria are still available ◮ What about signatures? 2
Gröbner bases ◮ Valuable tool for many questions related to polynomial equations (solving, elimination, dimension of the solutions...) ◮ Classically used for polynomials over fields ◮ Some applications with coefficients in general rings (elimination, combinatorics...) Many algorithms for fields ◮ First algorithm: Buchberger (1965) ◮ Optimizations related to selection strategies: “Normal” (1985), “Sugar” (1991) ◮ Criteria: Buchberger’s coprime and chain criteria (1979), Gebauer-Möller (1988) ◮ Replace polynomial arithmetic with linear algebra: Lazard (1983), F4 (1999) ◮ Signature-based criteria: F5 (2002), GVW (2010)... And for rings: ◮ Möller (1988) for general rings and principal domains, Kandri-Rodi Kapur (1988) for Euclidean domains... ◮ Optimizations and general criteria are still available ◮ What about signatures? This work : signature-based algorithms for PIDs 2
Outline 1. Reminders about Gröbner bases over fields ◮ Gröbner bases, Buchberger’s algorithm ◮ Signatures 2. Algorithms for rings ◮ Adding signatures to Möller’s weak GB algorithm ◮ Adding signatures to Möller’s strong GB algorithm 3. Proofs and experiments ◮ Skeleton of the proofs ◮ Experimental data ◮ Future work 3
Outline 1. Reminders about Gröbner bases over fields ◮ Gröbner bases, Buchberger’s algorithm ◮ Signatures 2. Algorithms for rings ◮ Adding signatures to Möller’s weak GB algorithm ◮ Adding signatures to Möller’s strong GB algorithm 3. Proofs and experiments ◮ Skeleton of the proofs ◮ Experimental data ◮ Future work 4
Gröbner bases: definitions ( R is a Noetherian ring) Definition (Leading term, monomial, coefficient) R ring, A = R [ X 1 , . . . , X n ] with a monomial order < , f = � a i X b i ◮ Leading term LT ( f ) = a i X b i with X b i > X b j if j � = i ◮ Leading monomial LM ( f ) = X b i ◮ Leading coefficient LC ( f ) = a i Definition (Weak/strong Gröbner basis) G ⊂ a = � f 1 , . . . , f n � ◮ G is a weak Gröbner basis ⇐ ⇒ � LT ( f ) : f ∈ a � = � LT ( g ) : g ∈ G � ◮ G is a strong Gröbner basis ⇐ ⇒ for all f ∈ a , f reduces to 0 modulo G Equivalent if R is a field 5
Gröbner bases: basic constructions ( R is a field) f ∈ A = R [ X ] , G = { g 1 , . . . , g s } ⊂ A Definition (S-polynomial) T ( i ) = LT ( g i ) , T ( i , j ) = lcm ( LT ( g i ) , LT ( g j )) S-Pol ( g i , g j ) = T ( i , j ) T ( i ) g i − T ( i , j ) T ( j ) g j Definition (Reduction) If LT ( f ) = cX a LT ( g i ) , then f reduces to h = f − cX a g modulo G . We use the same word for the transitive closure of the relation. Buchberger’s criterion G is a (strong) Gröbner basis ⇐ ⇒ for all i , j ∈ { 1 , . . . , s } , S-Pol ( g i , g j ) reduces to 0 modulo G . 6
Buchberger’s algorithm ( R is a field) (Strong) S-polynomial: f 1 , . . . , f m S-Pol = T ( i , j ) LT ( g i ) g i − T ( i , j ) LT ( g j ) g j g i e 1 , . . . , e m S-pol Gröbner basis g j ∅ (Strong) reduction: = 0 f � h = f − cX a LT ( g ) � = 0 Reduction S ( i , j ) S ( i , j ) S ( i , j ) 7
Computing in the free module ( R is a field) ◮ 1 st idea: keep track of the representation g = � i q i f i for g ∈ � f 1 , . . . , f m � [Möller, Mora, Traverso 1992] ◮ Work in the module A m = A e 1 ⊕ · · · ⊕ A e m with ¯ · : e i �→ ¯ e i = f i ◮ Example: S-polynomial: S-Pol ( g i , g j ) = T ( i , j ) T ( i ) g i − T ( i , j ) T ( j ) g j ◮ This computation is expensive! ◮ 2 nd idea: we don’t need the full representation, the largest term might be enough! [Faugère 2002 ; Gao, Volny, Wang 2010 ; Arri, Perry 2011... Eder, Faugère 2017] ◮ Define a signature s ( g ) of g as m m q i e i ∈ A m with ¯ � � s ( g ) = LT ( q j ) e j = LT ( g ) for some g = g = g = q i f i i = 1 i = 1 where q j is the last coef. � = 0 8
Signatures ( R is a field) ◮ Signatures are ordered as “position over term”: aX b e i < a ′ X b ′ e j ⇐ ⇒ i < j or i = j and X b < X b ′ ◮ Example: S-polynomial: S-Pol ( g i , g j ) = T ( i , j ) T ( i ) g i − T ( i , j ) T ( j ) g j Up to permutation, two situations: ◮ T ( i , j ) T ( i ) LT ( g i ) > T ( i , j ) LT ( S-Pol ( g i , g j )) = T ( i , j ) T ( j ) LT ( g j ) → T ( i ) LT ( g i ) ◮ T ( i , j ) T ( i ) LT ( g i ) ≃ T ( i , j ) LT ( S-Pol ( g i , g j )) ≤ T ( i , j ) T ( j ) LT ( g j ) → T ( i ) LT ( g i ) 9
Signatures ( R is a field) ◮ Signatures are ordered as “position over term”: aX b e i < a ′ X b ′ e j ⇐ ⇒ i < j or i = j and X b < X b ′ ◮ Example: S-polynomial: S-Pol ( g i , g j ) = T ( i , j ) T ( i ) g i − T ( i , j ) T ( j ) g j Up to permutation, two situations: ◮ T ( i , j ) T ( i ) s ( g i ) > T ( i , j ) s ( S-Pol ( g i , g j )) = T ( i , j ) T ( j ) s ( g j ) → T ( i ) s ( g i ) Regular S-polynomial ◮ T ( i , j ) T ( i ) s ( g i ) ≃ T ( i , j ) s ( S-Pol ( g i , g j )) ≤ T ( i , j ) T ( j ) s ( g j ) → T ( i ) s ( g i ) Non regular S-polynomial: possible signature drop ◮ Keeping track of the signature is free if we restrict to regular S-pols and reductions! 9
s -reductions, s -Gröbner bases ( R is a field) Definition (Signature reductions) f , g , h ∈ � f 1 , . . . , f m � with signatures, such that f reduces to h = f − cX a g The reduction is ◮ a s -reduction if X a s ( g ) ≤ s ( f ) ( i.e. s ( h ) ≤ s ( f ) ) ◮ a regular s -reduction if X a s ( g ) � s ( f ) ( i.e. s ( h ) = s ( f ) ) Definition (Signature Gröbner basis) G = { g 1 , . . . , g s } ⊂ a = � f 1 , . . . , f m � is a (strong) s -Gröbner basis iff for all f ∈ a , f s -reduces to 0 modulo G . Key theorem ◮ A s -Gröbner basis is a Gröbner basis ◮ Every ideal admits a finite s -Gröbner basis 10
Buchberger’s algorithm, with signatures ( R is a field) (Strong) S-polynomial: f 1 , . . . , f m S-Pol = T ( i , j ) LT ( g i ) g i − T ( i , j ) LT ( g j ) g j g i , s ( g i ) e 1 , . . . , e m Regular: T ( i , j ) LT ( g i ) s ( g i ) > T ( i , j ) LT ( g j ) s ( g j ) S-pol s -Gröbner basis g j S ( i , j ) = T ( i , j ) LT ( g i ) s ( g i ) ∅ (Strong) reduction: = 0 f � h = f − cX a LT ( g ) � = 0 Regular S ( i , j ) S ( i , j ) S ( i , j ) Regular: s ( f ) > X a s ( g ) reduction s ( h ) = s ( f ) 11
Consequences of signatures ( R is a field) Key property Buchberger’s algorithm with signatures computes GB elements with increasing signatures. Main consequence Buchberger’s algorithm with signatures is correct and computes a signature GB. Then we can add criteria... Singular criterion: eliminate some redundant computations If s ( g ) ≃ s ( g ′ ) then afer regular reduction, LM ( g ) = LM ( g ′ ) . F5 criterion: eliminate Koszul syzygies f i f j − f j f i = 0 If s ( g ) = LT ( g ′ ) e j and s ( g ′ ) = ⋆ e i for some indices i < j , then g reduces to 0 modulo the already computed basis. 12
Outline 1. Reminders about Gröbner bases over fields ◮ Gröbner bases, Buchberger’s algorithm ◮ Signatures 2. Algorithms for rings ◮ Adding signatures to Möller’s weak GB algorithm ◮ Adding signatures to Möller’s strong GB algorithm 3. Proofs and experiments ◮ Skeleton of the proofs ◮ Experimental data ◮ Future work 13
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