Circles in a Circle 1923 Oil on canvas 38 7/8 x 37 5/8 inches (98.7 x 95.6 cm) Wassily Kandinsky

Transcription

1 Perturbations of Roots under Linear Transformations of Polynomials Branko Ćurgus Western Washington University Bellingham, WA, USA February 19,

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3 Circles in a Circle 1923 Oil on canvas 38 7/8 x 37 5/8 inches (98.7 x 95.6 cm) Wassily Kandinsky Russian, worked in Germany and France lived

4 Philadelphia Museum of Art: The Louise and Walter Arensberg Collection

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6 Q. I. Rahman, G. Schmeisser: Analytic theory of polynomials. Oxford University Press, T. Sheil-Small: Complex polynomials. Cambridge University Press, M. Marden: Geometry of polynomials. Second edition, American Mathematical Society,

7 This is joint research with Vania Mascioni. On the location of critical points of polynomials. Proc. Amer. Math. Soc. 131 (2003), A contraction of the Lucas polygon. Proc. Amer. Math. Soc. 132 (2004), Roots and polynomials as homeomorphic spaces. Expositiones Mathematicae 24 (2006), Results of this talk are from a paper accepted in Constructive Approximation. 3

8 We study polynomials with complex coefficients a j : p(z) = a 0 + a 1 z a n z n

9 We study polynomials with complex coefficients a j : p(z) = a 0 + a 1 z a n z n P n = { all polynomials of degree n } 4

10 p Pn n o Z(p) = w C : p(w) = 0

11 p P n Z(p) = { w C : p(w) = 0 } L(P n ) = { all linear operators on P n }

12 p P n Z(p) = { w C : p(w) = 0 } L(P n ) = { all linear operators on P n } Vladimir Tulovsky: On perturbations of roots of polynomials. J. Analyse Math. 54 (1990),

13 Let T L(P n ). Z(p) = Z(T p) for all non-constant p P n if and only if T?

14 Let T L(P n ). Z(p) = Z(T p) for all non-constant p P n if and only if T = αi, α C \ {0} 6

15 Let T L(P n ). Z(p) Z(T p) for all non-constant p P n if and only if T?

16 Let T L(P n ). Z(p) Z(T p) for all non-constant p P n if and only if T = αi, α C \ {0} 7

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18 Let T L(P n ) and D(r) = { z C : z r }. C T > 0 such that ( Z(p) + D(CT ) ) Z(T p) for all non-constant p P n if and only if T? 8

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20 What is a simple example of such an operator?

21 What is a simple example of such an operator? (S α p)(z) = p(α + z)

22 What is a simple example of such an operator? (S α p)(z) = p(α + z) Z(S α p) = { α} + Z(p).

23 What is a simple example of such an operator? (S α p)(z) = p(α + z) Z(S α p) = { α} + Z(p). (S α p)(z) = p(z) + α 1! p (z) + + αn n! p(n) (z)

24 What is a simple example of such an operator? (S α p)(z) = p(α + z) Z(S α p) = { α} + Z(p). (S α p)(z) = p(z) + α 1! p (z) + + αn n! p(n) (z) S α = I + α 1! D + + αn n! Dn 9

25 Let T L(P n ) and D(r) = { z C : z r }. C T > 0 such that ( Z(p) + D(CT ) ) Z(T p) for all non-constant p P n if and only if

26 Let T L(P n ) and D(r) = { z C : z r }. C T > 0 such that ( Z(p) + D(CT ) ) Z(T p) for all non-constant p P n if and only if T = α 0 I + α 1 D + + α n D n, α

27 More than ( Z(p) + D(CT ) ) Z(T p) is true.

28 More than ( Z(p) + D(CT ) ) Z(T p) is true. T = α 0 I + α 1 D + + α n D n, α 0 0.

29 More than ( Z(p) + D(CT ) ) Z(T p) is true. T = α 0 I + α 1 D + + α n D n, α 0 0. Let φ n (z) = z n.

30 More than ( Z(p) + D(CT ) ) Z(T p) is true. T = α 0 I + α 1 D + + α n D n, α 0 0. Let φ n (z) = z n. Let K T = max { u : u Z(T φ n ) }.

31 More than ( Z(p) + D(CT ) ) Z(T p) is true. T = α 0 I + α 1 D + + α n D n, α 0 0. Let φ n (z) = z n. Let K T = max { u : u Z(T φ n ) }. Then Z(T p) Z(p) + D(K T ) for all non-constant p P n. 11

32 Let U and V be finite subsets of C. The Hausdorff distance is defined by: d H (U, V ) = min { r > 0 : V U + D(r), U V + D(r) }.

33 Let U and V be finite subsets of C. The Hausdorff distance is defined by: d H (U, V ) = min { r > 0 : V U + D(r), U V + D(r) }. Let T = α 0 I + α 1 D + + α n D n, α 0 0.

34 Let U and V be finite subsets of C. The Hausdorff distance is defined by: d H (U, V ) = min { r > 0 : V U + D(r), U V + D(r) }. Let T = α 0 I + α 1 D + + α n D n, α 0 0. Then d H ( Z(p), Z(T p) ) max { KT, K T 1 } for all non-constant p P n. 12

35 Let m N. Let Π m be the set of all permutations of {1,..., m}.

36 Let m N. Let Π m be the set of all permutations of {1,..., m}. Let U = {u 1,..., u m } and V complex numbers. = {v 1,..., v m } be multisets of m

37 Let m N. Let Π m be the set of all permutations of {1,..., m}. Let U = {u 1,..., u m } and V complex numbers. = {v 1,..., v m } be multisets of m The Fréchet distance is defined by: d F (U, V ) := min σ Π m max 1 k m u k v σ(k). 13

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39 Let T = α 0 I + α 1 D + + α n D n, α 0 0. Let γ 1,..., γ n be the roots of α 0 z n + α 1 z n α n 1 z + α n counted according to their multiplicities.

40 Let T = α 0 I + α 1 D + + α n D n, α 0 0. Let γ 1,..., γ n be the roots of α 0 z n + α 1 z n α n 1 z + α n counted according to their multiplicities. Then d F ( Z(p), Z(T p) ) n 2 ( γ γ n ) for all non-constant p P n. 14

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66 Let p P n be a polynomial with n distinct roots. Then Z(p) Z(p ) =. Define τ(p) = min { w v : w Z(p), v Z(p ) }

67 Let p P n be a polynomial with n distinct roots. Then Z(p) Z(p ) =. Define τ(p) = min { w v : w Z(p), v Z(p ) } c(p) = 1 n w Z(p) w

68 Let p P n be a polynomial with n distinct roots. Then Z(p) Z(p ) =. Define τ(p) = min { w v : w Z(p), v Z(p ) } c(p) = 1 n w Z(p) w ρ(p) = max { w c(p) : w Z(p) }

69 Let p P n be a polynomial with n distinct roots. Then Z(p) Z(p ) =. Define τ(p) = min { w v : w Z(p), v Z(p ) } c(p) = 1 n w Z(p) w ρ(p) = max { w c(p) : w Z(p) } spr(p) = τ(p) ( τ(p) ρ(p) ) n 2 15

70 A trivial example: Let r > 0. p(z) = z n r n, p (z) = nz n 1

71 A trivial example: Let r > 0. p(z) = z n r n, p (z) = nz n 1 Z(p) {w C : w = r}, Z(p ) = {0}

72 A trivial example: Let r > 0. p(z) = z n r n, p (z) = nz n 1 Z(p) {w C : w = r}, Z(p ) = {0} τ(p) = r, c(p) = 0, ρ(p) = r

73 A trivial example: Let r > 0. p(z) = z n r n, p (z) = nz n 1 Z(p) {w C : w = r}, Z(p ) = {0} τ(p) = r, c(p) = 0, ρ(p) = r spr(p) = r ( r r ) n 2 = r 16

74 A generalization: Let t > 0. Define H t L(P n ) by (H t p)(z) = p(z/t), p P n.

75 A generalization: Let t > 0. Define H t L(P n ) by (H t p)(z) = p(z/t), p P n. Then spr(h t p) = t spr(p). 17

76 Recall S α = I + αd + α2 2! D2 + + αn n! Dn.

77 Recall S α = I + αd + α2 2! D2 + + αn n! Dn. Let T = I + αd + α 2 D α n D n.

78 Recall S α = I + αd + α2 2! D2 + + αn n! Dn. Let T = I + αd + α 2 D α n D n. Then there exists a constant Γ T > 0 such that d F ( Z(Sα p), Z(T p) ) Γ T spr(p) for all p P n with n distinct roots. 18

79 Recall S α = I + αd + α2 2! D2 + + αn n! Dn. Let T = I + αd + α 2 D α n D n.

80 Recall S α = I + αd + α2 2! D2 + + αn n! Dn. Let T = I + αd + α 2 D α n D n. A corollary:

81 Recall S α = I + αd + α2 2! D2 + + αn n! Dn. Let T = I + αd + α 2 D α n D n. A corollary: For an arbitrary p P n with n distinct roots: lim d ( F Z(Sα H t p), Z(T H t p) ) = 0 t + 19