Signature-based Mllers algorithm for strong Grbner bases over PIDs - PowerPoint PPT Presentation

Signature-based Möller’s algorithm for strong Gröbner bases over PIDs Maria Francis 1 , Thibaut Verron 2 1. Indian Institute of Technology Hyderabad, Hyderabad, India 2. Institute for Algebra, Johannes Kepler University, Linz, Austria SIAM Conference on Applied Algebraic Geometry Mini-symposium Algebraic methods for polynomial system solving 13 July 2019, University of Bern, Switzerland 1

Gröbner bases X a = X a 1 1 · · · X a n n ◮ 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...) ◮ Z [ X 1 , . . . , X n ] is a central object in algebraic geometry Leading term, monomial, coefficient: R ring, A = R [ X 1 , . . . , X n ] with a monomial order < LT ( f ) c · X a f = + smaller terms LC ( f ) LM ( f ) Definition (Weak/strong Gröbner basis) G ⊂ I = � f 1 , . . . , f m � ◮ G is a weak Gröbner basis ⇐ ⇒ � LT ( f ) : f ∈ I � = � LT ( g ) : g ∈ G � ◮ G is a strong Gröbner basis ⇐ ⇒ for all f ∈ I , f reduces to 0 modulo G Equivalent if R is a field 2

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 S ( i , j ) S ( i , j ) ∅ (Strong) reduction: = 0 � = 0 f � h = f − c X a LT ( g ) Reduction 3

Why signatures? Problem: useless and redundant computations Example with a S-polynomial p = p 1 f 1 + p 2 f 2 + · · · + p k f k + · · · + p m f m q = q 1 f 1 + q 2 f 2 + · · · + q l f l + · · · + q m f m p = p 1 e 1 + p 2 e 2 + · · · + p k e k + · · · + p m e m q = q 1 e 1 + q 2 e 2 + · · · + q l e l + · · · + q m e m = LT ( p k ) e k + smaller terms = LT ( q l ) e l + smaller terms � s ( p ) = signature of p i < j “Position over term”: X a e i < X b e j if or i = j and X a < X b S-Pol ( p , q ) = µ p − ν q S-Pol ( p , q ) = µ ( p 1 e 1 + · · · + p k e k + · · · + p m e m ) − ν ( q 1 e 1 + · · · + q l e l + · · · + q m e m ) = µ LT ( p k ) e k − ν LT ( q l ) e l + smaller terms = µ LT ( p k ) e k + smaller terms if µ LT ( p k ) � ν LT ( q l ) e l Regular S-polynomial 4

Why signatures? Problem: useless and redundant computations ◮ 1 st idea: keep track of the representation of the ideal elements [Möller, Mora, Traverso 1992] Example with a S-polynomial p = p 1 f 1 + p 2 f 2 + · · · + p k f k + · · · + p m f m q = q 1 f 1 + q 2 f 2 + · · · + q l f l + · · · + q m f m p = p 1 e 1 + p 2 e 2 + · · · + p k e k + · · · + p m e m q = q 1 e 1 + q 2 e 2 + · · · + q l e l + · · · + q m e m = LT ( p k ) e k + smaller terms = LT ( q l ) e l + smaller terms � s ( p ) = signature of p i < j “Position over term”: X a e i < X b e j if or i = j and X a < X b S-Pol ( p , q ) = µ p − ν q S-Pol ( p , q ) = µ ( p 1 e 1 + · · · + p k e k + · · · + p m e m ) − ν ( q 1 e 1 + · · · + q l e l + · · · + q m e m ) = µ LT ( p k ) e k − ν LT ( q l ) e l + smaller terms = µ LT ( p k ) e k + smaller terms if µ LT ( p k ) � ν LT ( q l ) e l Regular S-polynomial 4

Why signatures? Problem: useless and redundant computations ◮ 1 st idea: keep track of the representation of the ideal elements [Möller, Mora, Traverso 1992] ◮ 2 nd idea: we do not need the full representation, the largest term is enough [Faugère 2002 ; Gao, Volny, Wang 2010 ; Arri, Perry 2011... Eder, Faugère 2017] Example with a S-polynomial p = p 1 f 1 + p 2 f 2 + · · · + p k f k + · · · + 0 f m q = q 1 f 1 + q 2 f 2 + · · · + q l f l + · · · + 0 f m p = p 1 e 1 + p 2 e 2 + · · · + p k e k + · · · + 0 e m q = q 1 e 1 + q 2 e 2 + · · · + q l e l + · · · + 0 e m = LT ( p k ) e k + smaller terms = LT ( q l ) e l + smaller terms � s ( p ) = signature of p i < j “Position over term”: X a e i < X b e j if or i = j and X a < X b S-Pol ( p , q ) = µ p − ν q S-Pol ( p , q ) = µ ( p 1 e 1 + · · · + p k e k + · · · + 0 e m ) − ν ( q 1 e 1 + · · · + q l e l + · · · + 0 e m ) = µ LT ( p k ) e k − ν LT ( q l ) e l + smaller terms = µ LT ( p k ) e k + smaller terms if µ LT ( p k ) � ν LT ( q l ) e l Regular S-polynomial 4

Why signatures? Problem: useless and redundant computations ◮ 1 st idea: keep track of the representation of the ideal elements [Möller, Mora, Traverso 1992] ◮ 2 nd idea: we do not need the full representation, the largest term is enough [Faugère 2002 ; Gao, Volny, Wang 2010 ; Arri, Perry 2011... Eder, Faugère 2017] Example with a S-polynomial p = p 1 f 1 + p 2 f 2 + · · · + p k f k + · · · + 0 f m q = q 1 f 1 + q 2 f 2 + · · · + q l f l + · · · + 0 f m p = p 1 e 1 + p 2 e 2 + · · · + p k e k + · · · + 0 e m q = q 1 e 1 + q 2 e 2 + · · · + q l e l + · · · + 0 e m = LT ( p k ) e k + smaller terms = LT ( q l ) e l + smaller terms � s ( p ) = signature of p i < j “Position over term”: X a e i < X b e j if or i = j and X a < X b S-Pol ( p , q ) = µ p − ν q S-Pol ( p , q ) = µ ( p 1 e 1 + · · · + p k e k + · · · + 0 e m ) − ν ( q 1 e 1 + · · · + q l e l + · · · + 0 e m ) = µ LT ( p k ) e k − ν LT ( q l ) e l + smaller terms = µ LT ( p k ) e k + smaller terms if µ LT ( p k ) � ν LT ( q l ) e l Regular S-polynomial 4

Why signatures? Problem: useless and redundant computations ◮ 1 st idea: keep track of the representation of the ideal elements [Möller, Mora, Traverso 1992] ◮ 2 nd idea: we do not need the full representation, the largest term is enough [Faugère 2002 ; Gao, Volny, Wang 2010 ; Arri, Perry 2011... Eder, Faugère 2017] Example with a S-polynomial p = p 1 f 1 + p 2 f 2 + · · · + p k f k + · · · + 0 f m q = q 1 f 1 + q 2 f 2 + · · · + q l f l + · · · + 0 f m p = p 1 e 1 + p 2 e 2 + · · · + p k e k + · · · + 0 e m q = q 1 e 1 + q 2 e 2 + · · · + q l e l + · · · + 0 e m = LT ( p k ) e k + smaller terms = LT ( q l ) e l + smaller terms � s ( p ) = signature of p i < j “Position over term”: X a e i < X b e j if or i = j and X a < X b S-Pol ( p , q ) = µ p − ν q S-Pol ( p , q ) = µ ( p 1 e 1 + · · · + p k e k + · · · + 0 e m ) − ν ( q 1 e 1 + · · · + q l e l + · · · + 0 e m ) = µ LT ( p k ) e k − ν LT ( q l ) e l + smaller terms = µ LT ( p k ) e k + smaller terms if µ LT ( p k ) � ν LT ( q l ) e l Regular S-polynomial 4

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 ) S ( i , j ) S ( i , j ) ∅ (Strong) reduction: = 0 � = 0 f � h = f − c X a LT ( g ) Regular Regular: s ( f ) > X a s ( g ) reduction s ( h ) = s ( f ) 5

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! 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. 6

Context and main results: what about rings? Buchberger (1965) Coefficients Faugère: F4 (1999) can be thrown away . Field . . Coefficients Euclidean ring Kandri-Rody, Kapur (1988) can be ordered Möller weak (1988) Coefficients Möller strong (1988) must be kept Principal ideal domain Lichtblau (2012) but can be ignored General (Noetherian) ring Möller weak (1988) Main question with signatures: how to order the coefficients of the signatures? With a total order, signature drops cannot be avoided [Eder, Popescu 2017] But with a partial order, signatures cannot decrease [Francis, V. 2018] (weak) [Francis, V. 2019] (strong) 7

Context and main results: what about rings? Buchberger (1965) Coefficients Faugère: F4 (1999) can be thrown away . Field . . Coefficients Euclidean ring Kandri-Rody, Kapur (1988) can be ordered Möller weak (1988) Coefficients Möller strong (1988) must be kept Principal ideal domain Lichtblau (2012) but can be ignored General (Noetherian) ring Möller weak (1988) Main question with signatures: how to order the coefficients of the signatures? With a total order, signature drops cannot be avoided [Eder, Popescu 2017] But with a partial order, signatures cannot decrease [Francis, V. 2018] (weak) [Francis, V. 2019] (strong) 7

Context and main results: what about rings? Buchberger (1965) Coefficients Faugère: F4 (1999) can be thrown away . Field . . Coefficients Euclidean ring Kandri-Rody, Kapur (1988) can be ordered Möller weak (1988) Coefficients Möller strong (1988) must be kept Principal ideal domain Lichtblau (2012) but can be ignored General (Noetherian) ring Möller weak (1988) Main question with signatures: how to order the coefficients of the signatures? With a total order, signature drops cannot be avoided [Eder, Popescu 2017] But with a partial order, signatures cannot decrease [Francis, V. 2018] (weak) [Francis, V. 2019] (strong) 7