On the Complexity of Computing Real Radicals of Polynomial Systems Mohab Safey El Din 1 Zhi-Hong Yang 2 Lihong Zhi 2 1 Sorbonne Universit´ e, CNRS , INRIA , Laboratoire d’Informatique de Paris 6, LIP6 , ´ Equipe PolSys 2 Key Lab of Mathematics Mechanization, Academy of Mathematics and Systems Science, CAS, China ISSAC’18, New York, July 16-19
Motivation Polynomial system solving over the reals: f = ( f 1 , . . . , f s ) ⊂ Q [ X 1 , . . . , X n ] V R ( f ) = { x ∈ R n | f 1 ( x ) = 0 , . . . , f s ( x ) = 0 } V V ◮ Numeric computation − → reliability issues, especially in the singular case. � ◮ Algebraic computation − → ( V = V V V C ( f ) ← → � f � ) What if V ∩ R n ⊂ Sing ( V ) ? x 2 1 + x 2 2 = 0 ւ ց • Dimensions are different. • (0 , 0) ∈ V , singular V V V V C ( f ) : lines V V R ( f ) : point • (0 , 0) ∈ V ∩ R n , smooth! � x 1 + ix 2 = 0 x 1 = x 2 = 0 x 1 − ix 2 = 0 2/15 ,
Example 1 [Everett, Lazard, Lazard, Safey El Din, 2007] Vor1 =( α 2 + β 2 + 1) a 2 λ 4 − 2 a (2 aβ 2 + ayβ + aαx − βα + 2 a + 2 aα 2 − βαa 2 ) λ 3 + ( β 2 + 6 a 2 β 2 − 2 βxa 3 − 6 βαa 3 + 6 yβa 2 − 6 aβα − 2 aβx + 6 αxa 2 + y 2 a 2 − 2 aαy + x 2 a 2 − 2 yαa 3 + 6 a 2 α 2 + a 4 α 2 + 4 a 2 ) λ 2 − 2( xa − ya 2 − 2 βa 2 − β + 2 aα + αa 3 )( xa − y − β + aα ) λ + (1 + a 2 )( xa − y − β + aα ) 2 . 3/15 ,
Example 1 [Everett, Lazard, Lazard, Safey El Din, 2007] Vor1 =( α 2 + β 2 + 1) a 2 λ 4 − 2 a (2 aβ 2 + ayβ + aαx − βα + 2 a + 2 aα 2 − βαa 2 ) λ 3 + ( β 2 + 6 a 2 β 2 − 2 βxa 3 − 6 βαa 3 + 6 yβa 2 − 6 aβα − 2 aβx + 6 αxa 2 + y 2 a 2 − 2 aαy + x 2 a 2 − 2 yαa 3 + 6 a 2 α 2 + a 4 α 2 + 4 a 2 ) λ 2 − 2( xa − ya 2 − 2 βa 2 − β + 2 aα + αa 3 )( xa − y − β + aα ) λ + (1 + a 2 )( xa − y − β + aα ) 2 . ◮ Real zeros of Vor1 are union of: � � � aα − ax + β − y = 0 aα + ax − β − y = 0 2 βλ + β + y = 0 λ + 1 = 0 λ = 0 a = 0 3/15 ,
Example 1 [Everett, Lazard, Lazard, Safey El Din, 2007] Vor1 =( α 2 + β 2 + 1) a 2 λ 4 − 2 a (2 aβ 2 + ayβ + aαx − βα + 2 a + 2 aα 2 − βαa 2 ) λ 3 + ( β 2 + 6 a 2 β 2 − 2 βxa 3 − 6 βαa 3 + 6 yβa 2 − 6 aβα − 2 aβx + 6 αxa 2 + y 2 a 2 − 2 aαy + x 2 a 2 − 2 yαa 3 + 6 a 2 α 2 + a 4 α 2 + 4 a 2 ) λ 2 − 2( xa − ya 2 − 2 βa 2 − β + 2 aα + αa 3 )( xa − y − β + aα ) λ + (1 + a 2 )( xa − y − β + aα ) 2 . ◮ Real zeros of Vor1 are union of: � � � aα − ax + β − y = 0 aα + ax − β − y = 0 2 βλ + β + y = 0 λ + 1 = 0 λ = 0 a = 0 ◮ Only one connected component, which is not easy to be seen from Vor1 . 3/15 ,
Problem ◮ f = ( f 1 , . . . , f s ) ⊂ Q [ X 1 , . . . X n ] . � re � f � : the vanishing ideal of V V V R ( f ) . ◮ √ ◮ An ideal I is called real if I = re I . ◮ D = max { deg f i , . . . , deg f s } . Input: f = ( f 1 , . . . , f s ) � Output: irreducible components of re � f � : ◮ generators, or ◮ rational parametrizations. 4/15 ,
Example 1 (Continued) Vor1 =( α 2 + β 2 + 1) a 2 λ 4 − 2 a (2 aβ 2 + ayβ + aαx − βα + 2 a + 2 aα 2 − βαa 2 ) λ 3 + ( β 2 + 6 a 2 β 2 − 2 βxa 3 − 6 βαa 3 + 6 yβa 2 − 6 aβα − 2 aβx + 6 αxa 2 + y 2 a 2 − 2 aαy + x 2 a 2 − 2 yαa 3 + 6 a 2 α 2 + a 4 α 2 + 4 a 2 ) λ 2 − 2( xa − ya 2 − 2 βa 2 − β + 2 aα + αa 3 )( xa − y − β + aα ) λ + (1 + a 2 )( xa − y − β + aα ) 2 . � Irreducible components of � Vor1 � : re P 1 = � aα − ax + β − y, λ + 1 � P 2 = � aα + ax − β − y, λ � P 3 = � 2 βλ + β + y, a � Timing: 9 sec. 5/15 ,
State of the art Exact computation: ◮ Becker, Neuhaus’1993, Neuhaus’1998, Spang’2007 Using Gr¨ obner bases to compute real radicals for arbitrary polynomial ideals. The complexity is D 2 O ( n 2) . Numerical approximations: ◮ Lasserre, Laurent, Rostalski’2008; Lasserre, Laurent, Mourrain, Rostalski, Tr´ ebuchet’2013 Using SDP relaxations to compute zero-dimensional real radical ideals. ◮ Ma, Wang, Zhi’2014 A certificate for computing real radicals using SDP relaxations. ◮ Brake, Hauenstein, Liddell’2016 A method based SDP programming for deciding if an ideal is real. 6/15 ,
Main Results f = ( f 1 , . . . , f s ) ⊂ Q [ X 1 , . . . , X n ] , r = dim � f � , D = max { deg f i } . D 2 O ( n 2) State of the art: Smooth case. A probabilistic algorithm computes generators of irreducible components of � f � using ( snD n ) O (1) operations in Q . � re General case. A probabilistic algorithm computes rational parametrizations of irreducible � f � using s O (1) ( nD ) O ( nr 2 r ) arithmetic operations in Q . � components of re 7/15 ,
Main idea Simple point criterion [Bochnak, Coste, Roy, 1998] � � ∂f i Prime I = � f 1 , . . . , f s � real ⇐ ⇒ ∃ x ∈ V V V R ( I ) s.t. rank ∂X j ( x ) = n − r , where r = dim I . 8/15 ,
Main idea Simple point criterion [Bochnak, Coste, Roy, 1998] � � ∂f i Prime I = � f 1 , . . . , f s � real ⇐ ⇒ ∃ x ∈ V V V R ( I ) s.t. rank ∂X j ( x ) = n − r , where r = dim I . Main idea: prime Yes f = ( f 1 , . . . , f s ) decomposition, P i real? P i real? output P i { P 1 , . . . , P m } No V singular locus of V V C ( f ) 8/15 ,
Singular point Singular point [Cox, Little, O’Shea, 1992] V ⊂ C n , p ∈ V , I I I ( V ) = � f 1 , . . . , f s � . The tangent space of V at p is s � n � ∂f j x ∈ C n � � � T p ( V ) = ∂X i ( p ) x i = 0 . � � j =1 i =1 dim p V = max { dim V i | p ∈ V i irreducible component of V } . ◮ Smooth Point: dim T p ( V ) = dim p V . ◮ Singular Point: dim T p ( V ) � = dim p V . ◮ Singular locus: Sing ( V ) = { p ∈ V | p is a singular point of V } . ◮ V is smooth if Sing ( V ) = ∅ . 9/15 ,
Smooth case V Input: f = ( f 1 , . . . , f s ) V V C ( f ) smooth sample points S sD O ( n ) Basu, Pollack, Roy, Rouillier, Safey El Din, Yes Output: 1 S = ∅ ? Schost, Bank, Giusti, Heintz No { P 1 , . . . , P m } = irreducible ( snD n ) O (1) (by Chow forms) � components of � f � Jeronimo, Krick, Sabia, Sombra/ Blanco, Jeronimo, Solern´ o Yes V V V R ( P i ) = drop P i (smoothness → evaluation) ∅ ? No Output: all remaining P i 10/15 ,
General case V Drop the smoothness assumption on V = V V C ( f ) . Difficulties: it may happen ◮ V ∩ R n ⊂ Sing ( V ) ; ◮ ... or even worse, in the singular locus of Sing ( V ) ; 11/15 ,
General case V Drop the smoothness assumption on V = V V C ( f ) . Difficulties: it may happen ◮ V ∩ R n ⊂ Sing ( V ) ; ◮ ... or even worse, in the singular locus of Sing ( V ) ; Standard idea: lazy representations for equidimensional components of V . ◮ equations and inequations. ◮ Triangular set decompositions (Wu, Lazard, etc.) or rational parametrizations (Giusti, Heintz, Morais, Pardo, etc.) 11/15 ,
Rational parametrization An r -equidimensional variety V ⊂ C n is the Zariski closure of the projection of the following set to X = ( X 1 , . . . , X n ) : ∂w ( T ) = v i ( T ) , ∂w ( T ) w ( T ) = 0 , X i � = 0 ∂T r +1 ∂T r +1 where T = ( T 1 , . . . , T r +1 ) , i = 1 , . . . , n . A rational parametrization of V : ◮ ℓ = ( λ 1 , . . . , λ r +1 ) , generic linear combinations of X 1 , . . . , X n ◮ polynomials w, v 1 , . . . , v n ∈ Q [ T ] . Denote R = (( w, v 1 , . . . , v n ) , ℓ ) . 12/15 ,
General case snD nrO (1) Lecerf Input: f = ( f 1 , . . . , f s ) R 1 , . . . , R t Yes ( nD ) O ( n 2 r ) Output: (1) R i = (1) ? No Basu, Pollack, Roy, Rouillier, No ∂w i R i , R i real? δ O ( r ) Safey El Din, ∂T ri +1 Schost, Bank, Giusti, Heintz Yes save R i clean up Output: all remaining R i 13/15 ,
Implementation Combing: ◮ SINGULAR: operating ideals [Greuel, Pfister]. ◮ Maple: computing sample points RAGlib [Safey El Din] (uses FGb [Faug` ere] for computing Gr¨ obner bases). 14/15 ,
Implementation Combing: ◮ SINGULAR: operating ideals [Greuel, Pfister]. ◮ Maple: computing sample points RAGlib [Safey El Din] (uses FGb [Faug` ere] for computing Gr¨ obner bases). Examples beyond the reach of the SINGULAR library realrad [Spang]. ◮ (Homotopy-1) 7 variables, degree 7. [Chen, Davenport, May, Moreno Maza, Xia, Xiao] f 1 = x 3 y 2 + c 1 x 3 y + y 2 + c 2 x + c 3 , f 2 = c 4 x 4 y 2 − x 2 y + y + c 5 , f 3 = c 4 − 1 . Timimg: 1 sec. ◮ (Essential variety) 9 variables, degree 3 [Fløystad, Kileel, Ottaviani] M ∈ R 3 × 3 | det( M ) = 0 , 2( MM T ) M − tr( MM T ) M = 0 E = � � Timing: 800 sec. 14/15 ,
Thank you! 15/15 ,
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