Regular Chains under Linear Changes of Coordinates and Applications - PowerPoint PPT Presentation

Regular Chains under Linear Changes of Coordinates and Applications Parisa Alvandi, Changbo Chen, Amir Hashemi, Marc Moreno Maza Western University, Canada September 17, 2015

Plan 1 Why Change of Coordinates? 2 Linear Change of Coordinates for Regular Chains 3 Noether Normalization and Regular Chains 4 Aller ` erateurs de sat( T ) a la pˆ eche aux g´ en´ 5 Conclusion

Plan 1 Why Change of Coordinates? 2 Linear Change of Coordinates for Regular Chains 3 Noether Normalization and Regular Chains 4 Aller ` erateurs de sat( T ) a la pˆ eche aux g´ en´ 5 Conclusion

Motivation (1/2) Polynomial system solving is an important problem in both science and engineering One method for solving such systems relies on triangular decompositions A triangular decomposition encodes the solutions of a polynomial system using special sub-systems called regular chains. Several encodings are possible. Example For the variable order b < a < y < x , with F = { a x + b, b x + y } , we have the following Z , V ( F ) = V ( T 1 ) \ V ( a b ) with T 1 = { b x + y, a y − b 2 } or V ( F ) = ( V ( T 1 ) \ V ( a b )) ∪ V ( T 2 ) ∪ V ( T 3 ) with T 2 = { x, y, b } and T 3 = { y, a, b } .

Motivation (2/2) Example (Cont’d) Recall F = { a x + b, b x + y } , T 1 = { b x + y, a y − b 2 } , T 2 = { x, y, b } and T 3 = { y, a, b } : Z implicitly describes the lines (0 , 0 , a, 0) and V ( F ) = V ( T 1 ) \ V ( a b ) ( x, 0 , 0 , 0) , whereas V ( F ) = V ( T 1 ) \ V ( t ) ∪ V ( T 2 ) ∪ V ( T 3 ) explicitly gives all points. Z = V ( T 1 : ( ab ) ∞ ) . Observe that we have V ( T 1 ) � = V ( T 1 ) \ V ( a b ) Question For F ⊂ Q [ x 1 , . . . , x n ] and a regular chain T ⊂ Q [ x 1 , . . . , x n ] with h T as Z how to product of initials such that we have V ( F ) = V ( T ) \ V ( h T ) compute V ( F ) \ ( V ( T ) \ V ( h T )) if only T (thus not F ) is known?

The problem: formal statement Notations Let T ⊂ C [ x 1 < · · · < x n ] be a regular chain. Let h T be the product of initials of polynomials of T . Let W ( T ) be the quasi-component of T , that is V ( T ) \ V ( h T ) . Z is the intersection of all algebraic sets containing W ( T ) . W ( T ) Problem statement Compute the non-trivial limit points of W ( T ) , that is, the set Z \ W ( T ) . lim( W ( T )) = W ( T ) Basic properties Z = V (sat( T )) where sat( T ) := � T � : h ∞ W ( T ) T , lim( W ( T )) = W ( T ) ∩ V ( h T ) , If dim(sat( T )) = d then lim( W ( T )) = ∅ or dim(lim( W ( T ))) = d − 1 .

Why is the problem difficult? Remark Z ⊆ V ( T ) but Given regular chain T ⊂ C [ x 1 < · · · < x n ] , we have W ( T ) Z � = V ( T ) W ( T ) may hold, which implies that a command like Triangularize( T ) may not Z , not even implicitly. compute W ( T ) Example Consider T = { z x − y 2 , y 4 − z 5 } . We have V ( T ) = W ( T ) ∪ V ( y, z ) Z = W ( T ) ∪ V ( y, z, x ) W ( T ) The former can be computed by Triangularize( T ) with output=lazard option while the latter requires to compute a generating set of Z . sat( T ) = T : h ∞ T since we have V (sat( T )) = W ( T )

Using Puiseux series In our CASC 2013 paper, we compute lim( W ( T )) whenever T is a one-dimensional regular chain over C : computations done w.r.t Euclidean topology (instead of Zariski topology) thanks to a theorem of D. Mumford. relies on Puiseux parametrizations not trivial to extend to a regular chain in higher dimension Example � x 1 x 2 3 + x 2 T := x 1 x 2 2 + x 2 + x 1 The regular chain T has four Puiseux expansions around x 1 = 0 : � x 3 � x 3 1 + O ( x 2 − 1 + O ( x 2 = 1 ) = 1 ) − x 1 + O ( x 2 − x 1 + O ( x 2 x 2 = 1 ) x 2 = 1 ) � x 3 � x 3 x 1 − 1 − 1 − x 1 − 1 + 1 2 x 1 + O ( x 2 2 x 1 + O ( x 2 = 1 ) = 1 ) − x 1 − 1 + x 1 + O ( x 2 − x 1 − 1 + x 1 + O ( x 2 x 2 = 1 ) x 2 = 1 )

Why using change of coordinate system? Motivation This is a fundamental technique to obtain a more convenient representation, and reveal properties, of the algebraic or differential representation of a geometrical object. Applications of random linear changes of coordinates ⊲ Obtaining a separating element, in computing rational univariate representation (RUR) of a zero-dimensional polynomial ideal. ⊲ Getting rid off “vertical components” for instance in computing the tangent cone of a space curve (see yesterday’s talk). ⊲ Noether normalization of a polynomial ideal. Our goals Compute lim( W ( T )) , as stated after, but also Study Noether normalization for ideals of the form sat( T ) .

How to use change of coordinates for computing lim( W ( T )) ? (1/2) First idea: Lever l’ind´ etermination Since W ( T ) = V ( T ) \ V ( h T ) , the difficulty in computing lim( W ( T )) is to “approach” V ( h T ) while staying in W ( T ) . Hence: Find a linear change of coordinates A and a regular chain C such that W A ( T ) = W ( C ) and we can converge to V A ( h T ) within W ( C ) (thus staying away of V ( h C ) ) � A − 1 � V ( C ) ∩ V A ( h T ) Then, we have lim( W ( T )) = Example Consider T := { x 4 , x 2 x 3 + x 2 1 } ⊂ Q [ x 1 < x 2 < x 3 < x 4 ] and the linear change of coordinates A : ( x 1 , x 2 , x 3 , x 4 ) �− → ( x 4 , x 2 + x 3 , x 2 , x 1 ) Using the PALGIE algorithm, we obtain C := { x 4 , x 2 3 + x 2 x 3 + x 2 1 } . Since C is monic, we can converge to V A ( h T ) within W ( C ) and have: � A − 1 � V ( C ) ∩ V A ( h T ) lim( W ( T )) = = V ( x 4 , x 2 , x 1 ) .

How to use change of coordinates for computing lim( W ( T )) ? (2/2) Second idea: Aller ` a la pˆ eche aux g´ en´ erateurs de sat( T ) Recall lim( W ( T )) = W ( T ) ∩ V ( h T ) ⊆ V ( T ) ∩ V ( h T ) Since W ( T ) = V (sat( T )) , there exist polynomial sets F ⊆ I ( V (sat( T ))) such that V ( T ∪ F ∪ h T ) = lim( W ( T )) holds. One may obtain such F by applying a change of coordinates A to T . Example Let T := { x 5 2 − x 4 1 , x 1 x 3 − x 2 2 } be a regular chain of Q [ x 1 < x 2 < x 3 ] . Let C := { x 5 3 − x 3 1 , x 2 3 x 2 − x 2 1 } be a regular chain of Q [ x 1 < x 3 < x 2 ] for which we have sat( C ) = sat( T ) . � � We shall exhibit a theorem implying � T, C � = sat( T ) from which we shall deduce lim( W ( T )) = V ( x 1 , x 2 , x 3 ) .

Plan 1 Why Change of Coordinates? 2 Linear Change of Coordinates for Regular Chains 3 Noether Normalization and Regular Chains 4 Aller ` erateurs de sat( T ) a la pˆ eche aux g´ en´ 5 Conclusion

Linear change of coordinates Notations Let k be a field and x = x 1 < · · · < x n be n ordered variables. Linear change of coordinates n any bijective map A of the form We call linear change of coordinates in k n n → A : k k (1) �− → ( A 1 ( x ) , . . . , A n ( x )) x where A 1 , . . . , A n are linear forms over k . Notation For f ∈ k [ x 1 , . . . , x n ] , we write f A ( x ) := f ( A 1 ( x ) , . . . , A n ( x )) . V A ( F ) := V ( { f A | f ∈ F } ) and W A ( T ) := V A ( T ) \ V A ( h T ) . For U := V ( F ) with F ⊂ k [ x 1 , . . . , x n ] , we define U A := V A ( F ) . For I := � F � , we define I A := � f A | f ∈ F � .

Change of Variable Order Problem 1 Given two orderings R 1 and R 2 on { x 1 , . . . , x n } , and T ⊂ k [ x ] a regular chain w.r.t R 1 , then compute finitely many regular chains C 1 , . . . , C e w.r.t R 2 such that Z = W ( C 1 ) Z ∪ · · · ∪ W ( C e ) Z W ( T )

Change of Variable Order Problem 1 Given two orderings R 1 and R 2 on { x 1 , . . . , x n } , and T ⊂ k [ x ] a regular chain w.r.t R 1 , then compute finitely many regular chains C 1 , . . . , C e w.r.t R 2 such that Z = W ( C 1 ) Z ∪ · · · ∪ W ( C e ) Z W ( T ) Example Let T = { z x 2 + y 2 , y 4 − z 3 } be a regular chain w.r.t R = z < y < x . Let R ′ = z < x < y , then C = PALGIE ( T, R ′ ) = { y 2 + x 2 z, x 4 − z } . In fact, we have sat( T ) R = sat( C ) R ′ .

Change of Variable Order Problem 1 Given two orderings R 1 and R 2 on { x 1 , . . . , x n } , and T ⊂ k [ x ] a regular chain w.r.t R 1 , then compute finitely many regular chains C 1 , . . . , C e w.r.t R 2 such that Z = W ( C 1 ) Z ∪ · · · ∪ W ( C e ) Z W ( T ) In the work of F. Boulier, F. Lemaire, and M. M. M. the differential counterpart of this problem, assuming sat( T ) is prime. An answer can be derived for the algebraic case and this algorithm is called PALGIE (Prime ALGebraic IdEal).

Change of Variable Order Problem 1 Given two orderings R 1 and R 2 on { x 1 , . . . , x n } , and T ⊂ k [ x ] a regular chain w.r.t R 1 , then compute finitely many regular chains C 1 , . . . , C e w.r.t R 2 such that Z = W ( C 1 ) Z ∪ · · · ∪ W ( C e ) Z W ( T ) Extending the PALGIE algorithm to a solution of the problem above can be achieved by standard methods from regular chains theory.

Change of Coordinate System Problem 2 Given a regular chain T and a linear change of coordinates A , compute finitely many regular chains C 1 , . . . , C e such that , Z = W ( C 1 ) Z ∪ · · · ∪ W ( C e ) Z W A ( T ) Given A is a linear change of coordinate system and T = { t 1 ( x 1 , . . . , x d ) , . . . , t n − d ( x 1 , . . . , x n ) } , Apply the extended version of PALGIE algorithm to  t A n − d ( x 1 , . . . , x n ) = 0   .  .  . T A t A 1 ( x 1 , . . . , x d ) = 0    h A T � = 0 

Plan 1 Why Change of Coordinates? 2 Linear Change of Coordinates for Regular Chains 3 Noether Normalization and Regular Chains 4 Aller ` erateurs de sat( T ) a la pˆ eche aux g´ en´ 5 Conclusion