Computing Grbner bases for quasi-homogeneous systems Jean-Charles - PowerPoint PPT Presentation

Computing Gröbner bases for quasi-homogeneous systems Jean-Charles Faugère 1 Mohab Safey El Din 12 Thibaut Verron 13 1 Université Pierre et Marie Curie, Paris 6, France INRIA Paris-Rocquencourt, Équipe P OL S YS Laboratoire d’Informatique de Paris 6, UMR CNRS 7606 2 Institut Universitaire de France 3 École Normale Supérieure March 22, 2013 Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 1

Motivations Parametrization of the solutions Polynomial system q ( T ) = 0 f 1 ( X ) = · · · = f m ( X ) = 0 Gröbner X = p ( T ) basis Applications: ◮ Cryptography ◮ Physics, industry... ◮ Theory (algo. geometry) Difficult problem Examples of successfully studied structures: ◮ NP-hard in finite field ◮ Homogeneous ◮ Exponential number of ◮ Bihomogeneous: solutions [ FSS10b ] ◮ Group symmetries: Problem : Exploit the structures of the e.g [ FS12 ] system ◮ Quasi-homogeneous Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 2

Motivations Parametrization of the solutions Polynomial system q ( T ) = 0 f 1 ( X ) = · · · = f m ( X ) = 0 Gröbner X = p ( T ) basis Row-echelon form of Applications: the Macaulay matrix ◮ Cryptography .   . ◮ Physics, industry... .   ◮ Theory (algo. m i F j     . geometry)   . . Difficult problem Examples of successfully studied structures: ◮ NP-hard in finite field ◮ Homogeneous ◮ Exponential number of ◮ Bihomogeneous: solutions [ FSS10b ] ◮ Group symmetries: Problem : Exploit the structures of the e.g [ FS12 ] system ◮ Quasi-homogeneous Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 2

Motivations Parametrization of the solutions Polynomial system q ( T ) = 0 f 1 ( X ) = · · · = f m ( X ) = 0 Gröbner X = p ( T ) basis Row-echelon form of Applications: the Macaulay matrix ◮ Cryptography .   . ◮ Physics, industry... .   ◮ Theory (algo. m i F j     . geometry)   . . Difficult problem Examples of successfully studied structures: ◮ NP-hard in finite field ◮ Homogeneous ◮ Exponential number of ◮ Bihomogeneous: solutions [ FSS10b ] ◮ Group symmetries: Problem : Exploit the structures of the e.g [ FS12 ] system ◮ Quasi-homogeneous Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 2

Motivations Parametrization of the solutions Polynomial system q ( T ) = 0 f 1 ( X ) = · · · = f m ( X ) = 0 Gröbner X = p ( T ) basis Row-echelon form of Applications: the Macaulay matrix ◮ Cryptography .   . ◮ Physics, industry... .   ◮ Theory (algo. m i F j     . geometry)   . . Difficult problem Examples of successfully studied structures: ◮ NP-hard in finite field ◮ Homogeneous ◮ Exponential number of ◮ Bihomogeneous: solutions [ FSS10b ] ◮ Group symmetries: Problem : Exploit the structures of the e.g [ FS12 ] system ◮ Quasi-homogeneous Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 2

Motivations Parametrization of the solutions Polynomial system q ( T ) = 0 f 1 ( X ) = · · · = f m ( X ) = 0 Gröbner X = p ( T ) basis Row-echelon form of Applications: the Macaulay matrix ◮ Cryptography .   . ◮ Physics, industry... .   ◮ Theory (algo. m i F j     . geometry)   . . Difficult problem Examples of successfully studied structures: ◮ NP-hard in finite field ◮ Homogeneous ◮ Exponential number of ◮ Bihomogeneous: solutions [ FSS10b ] ◮ Group symmetries: Problem : Exploit the structures of the e.g [ FS12 ] system ◮ Quasi-homogeneous Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 2

Definitions of quasi-homogeneous systems Definition System of weights: W = ( w 1 , . . . , w n ) ∈ N n Weighted degree: deg W ( X α 1 n ) = � n . . . X α n i = 1 w i α i 1 Quasi-homogeneous polynomial: poly. containing only monomials of same W -degree e.g. X 2 + XY 2 + Y 4 for W = ( 2 , 1 ) ◮ Homogeneous systems are W -homogeneous with weights ( 1 , . . . , 1 ) . Applications Physical system Polynomial inversion  Volume = Area × Height X = T 2 + U 2 Weight 2     Y = T 3 − TU 2 Weight 3 Weight 3 Weight 2 Weight 1 Weight 1  Z = T + 2 U            P ( X , Y , Z ) = 0 Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 3

Definitions of quasi-homogeneous systems Definition System of weights: W = ( w 1 , . . . , w n ) ∈ N n Weighted degree: deg W ( X α 1 n ) = � n . . . X α n i = 1 w i α i 1 Quasi-homogeneous polynomial: poly. containing only monomials of same W -degree e.g. X 2 + XY 2 + Y 4 for W = ( 2 , 1 ) ◮ Homogeneous systems are W -homogeneous with weights ( 1 , . . . , 1 ) . Applications Physical system Polynomial inversion  Volume = Area × Height X = T 2 + U 2 Weight 2     Y = T 3 − TU 2 Weight 3 Weight 3 Weight 2 Weight 1 Weight 1  Z = T + 2 U            P ( X , Y , Z ) = 0 Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 3

Usual two-steps strategy in the zero-dimensional case Parametrization Buchberger Change of ordering F 4 F 5 FGLM GR EV L EX Initial system L EX basis basis d max N ω nD ω d max Relevant complexity parameters ◮ d max = highest degree reached by F 5   N d Macaulay matrix ◮ Less than the degree of regularity d reg .   at degree d For generic homo. systems:                    n N d � ( d i − 1 ) + 1 [ Lazard83 ] d reg = For an homogeneous system: i = 1 ◮ D = degree of the ideal � � n + d − 1 N d = = number of solutions in dim. 0 d n � d i (homo. generic case) = i = 1 Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 4

Usual two-steps strategy in the zero-dimensional case Parametrization Change of ordering F 5 FGLM GR EV L EX Initial system L EX basis basis d max N ω nD ω d max Relevant complexity parameters ◮ d max = highest degree reached by F 5   N d Macaulay matrix ◮ Less than the degree of regularity d reg .   at degree d For generic homo. systems:                    n N d � ( d i − 1 ) + 1 [ Lazard83 ] d reg = For an homogeneous system: i = 1 ◮ D = degree of the ideal � � n + d − 1 N d = = number of solutions in dim. 0 d n � d i (homo. generic case) = i = 1 Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 4

Main results Adaptation of the usual strategy, so that we still have the complexity: ◮ C F 5 = O � � d reg N ω d reg ◮ C FGLM = O ( nD ω ) with estimations of the parameters for generic quasi-homogeneous systems: � n i = 1 d i ◮ D = � n i = 1 w i n � ◮ d reg = ( d i − w i ) + max { w j } i = 1 � � 1 n + d − 1 ◮ N d ≃ � n d i = 1 w i Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 5

Main results Adaptation of the usual strategy, so that we still have the complexity: ◮ C F 5 = O � � d reg N ω d reg ◮ C FGLM = O ( nD ω ) with estimations of the parameters for generic quasi-homogeneous systems: n � ◮ D = d i i = 1 n � ( d i − 1 ) + 1 ◮ d reg = i = 1 � � n + d − 1 ◮ N d = d Remark If we set the weights to ( 1 , . . . , 1 ) , we recover the usual values for homogeneous systems. Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 5

Setting a road-map Input ◮ W = ( w 1 , . . . , w n ) system of weights. ◮ F = ( f 1 , . . . , f n ) generic sequence of W -homogeneous polynomials with W -degree ( d 1 , . . . , d n ) . General road-map: 1. Find a generic property which rules out all reductions to zero 2. Design new algorithms to take advantage of this structure 3. Obtain complexity results Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 6

Setting a road-map Input ◮ W = ( w 1 , . . . , w n ) system of weights. ◮ F = ( f 1 , . . . , f n ) generic sequence of W -homogeneous polynomials with W -degree ( d 1 , . . . , d n ) . General road-map: 1. Find a generic property which rules out all reductions to zero ◮ Does the F 5 -criterion still work for quasi-homo. regular sequences? ◮ Are quasi-homo. regular sequences still generic? 2. Design new algorithms to take advantage of this structure 3. Obtain complexity results Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 6

Setting a road-map Input ◮ W = ( w 1 , . . . , w n ) system of weights. ◮ F = ( f 1 , . . . , f n ) generic sequence of W -homogeneous polynomials with W -degree ( d 1 , . . . , d n ) . General road-map: 1. Find a generic property which rules out all reductions to zero ◮ Does the F 5 -criterion still work for quasi-homo. regular sequences? ◮ Are quasi-homo. regular sequences still generic? 2. Design new algorithms to take advantage of this structure ◮ Adapt the matrix-F 5 algorithm to reduce the size of the computed matrices 3. Obtain complexity results Computing Gröbner bases for quasi-homogeneous systems 2013-03-22 6