New conditions for the intersection of algebraic curves with polydisk Yacine Bouzidi ⋆ ⋆ INRIA Lille - Nord Europe, ⋆ ②❛❝✐♥❡✳❜♦✉③✐❞✐❅✐♥r✐❛✳❢r Luminy, Marseille, February 4th, 2019 Journées Nationales de Calcul Formel Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 1 / 21
Intersection with polydisk: a historical problem Consider the closed unit polydisk of C n U n := { z = ( z 1 , . . . , z n ) ∈ C n | | z i | ≤ 1 , i = 1 , . . . , n } Let V ⊂ C n be an algebraic variety V := { z = ( z 1 , . . . , z n ) ∈ C n | f 1 ( z ) = · · · = f m ( z ) = 0 } , where f 1 , . . . , f m ∈ Q [ z 1 , . . . , z n ] Problem : Decide whether or not V ∩ U n = ∅ Application : m = 1: stability condition for n -dimensional systems m > 1: stabilization condition for n -dimensional systems Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 2 / 21
From a complex to a real problem For a polynomial system { f 1 , . . . , f m } ⊂ Q [ z 1 , . . . , z n ] If z k = a k + i b k , a k , b k ∈ R the problem is equivalent to: R ( f j ( a 1 + i b 1 , . . . , a n + i b n )) = 0 j = 1 , . . . , m C ( f j ( a 1 + i b 1 , . . . , a n + i b n )) = 0 a 2 k + b 2 k − 1 ≤ 0 k = 1 , . . . , n Problem reduced to testing the emptiness of a semi-algebraic set in R 2 n Generically of dimension 2 ( n − m ) Drawback : the number of variables is doubled ! Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 3 / 21
The Strintzis-Decarlo conditions For the case m = 1, simpler conditions have been derived Theorem (Strintzis, Decarlo et. al. 77) Let f ( z 1 , . . . , z n ) ∈ R [ z 1 , . . . , z n ] , the following two conditions are equivalent: f ( z 1 , . . . , z n ) � = 0 when | z 1 | ≤ 1 , . . . , | z n | ≤ 1 . f ( z 1 , 1 , . . . , 1 ) � = 0 , when | z 1 | ≤ 1 , f ( 1 , z 2 , 1 , . . . , 1 ) � = 0 , when | z 2 | ≤ 1 , . . . . . . f ( 1 , . . . , 1 , z n ) � = 0 , when | z n | ≤ 1 , f ( z 1 , . . . , z n ) � = 0 , when | z 1 | = . . . = | z n | = 1 . The problem reduces to an intersection with T n := { z = ( z 1 , . . . , z n ) ∈ C n | | z k | = 1 , k = 1 , . . . , n } Setting z i = x + i x − i in the last condition � checking the emptiness of an algebraic set in R n . [B. Quadrat, Rouillier 2016] Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 4 / 21
Idea of the original proof Consider w.lo.g the case n = 2 f ( 1 , z 2 ) � = 0 , when | z 2 | ≤ 1 f ( z 1 , z 2 ) � = 0 when | z 1 | ≤ 1 , | z 2 | ≤ 1 ⇐ ⇒ f ( z 1 , 1 ) � = 0 , when | z 1 | ≤ 1 f ( z 1 , z 2 ) � = 0 , when | z 1 | = | z 2 | = 1 Proof based on the continuity of the following function in U 1 � ∂ f ( z 1 , z 2 ) [ f ( z 1 , z 2 )] − 1 d z 2 N ( z 1 ) = 2 π j ∂ z 2 | z 2 | = 1 N ( z 1 ) is the number of zeros in z 2 of f ( z 1 , z 2 ) lying in | z 2 | ≤ 1 N ( z 1 ) is integer-valued � constant � 0 It does not generalize to arbitrary algebraic varieties ! at least straighforwardly. Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 5 / 21
Our contributions 1 New conditions for the case of complete intersection algebraic curves � The main ingredient : a new Strintzis-like theorem based on continuity arguments 2 A new algorithm for testing the intersection between algebraic curves and polydisk � The problem is reduced to zero-dimensional systems solving For simplicity, we focus in this talk on the case of algebraic curves in C 3 Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 6 / 21
The simplest case : algebraic curves in C 2 Let C be an algebraic curve in C 2 C := { ( z 1 , z 2 ) ∈ C 2 | f ( z 1 , z 2 ) = 0 } , where f ( z 1 , z 2 ) ∈ Q [ z 1 , z 2 ] Theorem (Strintzis, Decarlo et. al. 77) The following two conditions are equivalent f ( z 1 , z z ) � = 0 when | z 1 | ≤ 1 , | z 2 | ≤ 1 . f ( 1 , z 2 ) � = 0 , when | z 2 | ≤ 1 f ( z 1 , 1 ) � = 0 , when | z 1 | ≤ 1 f ( z 1 , z 2 ) � = 0 , when | z 1 | = | z 2 | = 1 Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 7 / 21
Sketch of the proof [B. and Moroz] We proceed by contraposition. Let ( α, β ) ∈ U 2 | f ( α, β ) = 0. Consider the continuous path inside the complex unit disk: [ 0 , 1 ] − → U α ( t ) : t �− → ( 1 − t ) α + t and consider the polynomial 2 + a n − 1 ( t ) z n − 1 f ( α ( t ) , z 2 ) = a n ( t ) z n + · · · + a 0 ( t ) 2 Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 8 / 21
Sketch of the proof [B. and Moroz] We proceed by contraposition. Let ( α, β ) ∈ U 2 | f ( α, β ) = 0. Consider the continuous path inside the complex unit disk: [ 0 , 1 ] − → U α ( t ) : t �− → ( 1 − t ) α + t and consider the polynomial 2 + a n − 1 ( t ) z n − 1 f ( α ( t ) , z 2 ) = a n ( t ) z n + · · · + a 0 ( t ) 2 Two cases: ∀ t ∈ [ 0 , 1 ] , a n ( t ) � = 0: the roots of f ( α ( t ) , z 2 ) vary continuously when t goes from 0 to 1 = ⇒ ∃ β 1 and β ( t ) : [ 0 , 1 ] → C such that f ( 1 , β 1 ) = 0 and β ( 0 ) = β, β ( 1 ) = β 1 . Two cases: | β 1 | ≤ 1 = ⇒ ∃ β ∈ U | f ( 1 , β ) = 0 (first condition of the theorem) | β 1 | > 1, by the continuity of the norm = ⇒ ∃ ( α, β ) ∈ U × T | f ( α, β ) = 0 Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 8 / 21
Sketch of the proof (end) ∃ t 0 ∈ [ 0 , 1 ] , a n ( t 0 ) = a n − 1 ( t 0 ) = · · · = a m + 1 ( t 0 ) = 0 , a m ( t 0 ) � = 0: n − m roots of f go continuously to infinity while t tends to t 0 . Two cases: ⇒ back to the first case β is not among these roots = β is among these roots, by the continuity of the norm = ⇒ ∃ t 1 ≤ t 0 and ( α 1 = α ( t 1 ) , β 1 = β ( t 1 )) ∈ U × T such that f ( α 1 , β 1 ) = 0 Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 9 / 21
Sketch of the proof (end) ∃ t 0 ∈ [ 0 , 1 ] , a n ( t 0 ) = a n − 1 ( t 0 ) = · · · = a m + 1 ( t 0 ) = 0 , a m ( t 0 ) � = 0: n − m roots of f go continuously to infinity while t tends to t 0 . Two cases: ⇒ back to the first case β is not among these roots = β is among these roots, by the continuity of the norm = ⇒ ∃ t 1 ≤ t 0 and ( α 1 = α ( t 1 ) , β 1 = β ( t 1 )) ∈ U × T such that f ( α 1 , β 1 ) = 0 One condition left, ∃ ( α, β ) ∈ U × T | f ( α, β ) = 0 We proceed in the same way considering this time the following continuous path in T [ 0 , 1 ] − → T β ( t ) : e i ( 1 − t ) θ t �− → where β = e i θ Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 9 / 21
Algebraic curves in C 3 Let C be an algebraic curve in C 3 C := { ( z 1 , z 2 , z 3 ) ∈ C 3 | f ( z 1 , z 2 , z 3 ) = g ( z 1 , z 2 , z 3 ) = 0 } , where f , g ∈ Q [ z 1 , z 2 , z 3 ] are in complete intersection. We make the following assumptions • The ideal � f , g � is radical • For any fixed value of z i , ♯ V C ( � f ( ., z i , . ) , g ( ., z i , . ) � ) < ∞ . � The curve C does not admit a whole component lying in the plan orthogonal to any direction z i . Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 10 / 21
Algebraic curves in C 3 For a given z i , consider the following canonical projection map C 3 − → C Π i : ( z 1 , z 2 , z 3 ) �− → z i As well as the following sets • V s ⊂ C 2 is the set of singular points of C = V ( � f , g � ) . c ⊂ C 2 is the set of critical points of Π i restricted to C and Π i ( V i • V i c ) its projection on the z i -axis. • V i ∞ ⊂ C is the set of non-properness points of Π i , i.e.,: z i ∈ C such that Π − 1 ( V ) ∩ C is not compact for any compact neighborhood V of z i . i Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 11 / 21
Algebraic curves in C 3 Theorem Under our assumptions, V i c ∪ V s and V i ∞ are Q -Zariski closed sets of dimension zero. Proof • Q -Zariski closedness c ∪ V s = V ( � f , g , Jac z i ( f , g ) � ) where Jac z i ( f , g ) = � ∂ f ∂ g ∂ f ∂ g V i ∂ z k − ∂ z j � with j , k � = i ∂ z j ∂ z k V i ∞ = π ( C p ∩ H ∞ ) where C p is the projective closure of C in C × P 2 , H ∞ is the plan at infinity in C × P 2 and π : C × P 2 → C the projection. • Zero-dimensionality Sard theorem + assumptions � V i c ∪ V s is zero-dimensional Non-properness locus is of co-dimension 1 � V i ∞ is zero-dimensional Yacine Bouzidi INRIA Lille, Projet NON-A Intersection of curves with polydisks February 4, 2019 12 / 21
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