Perturbations of Roots under Linear Transformations of Polynomials
Perturbations of Roots under Linear Transformations of Polynomials ´ urgus Branko C Western Washington University Bellingham, WA, USA February 19, 2006
1
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 1866 - 1944
Philadelphia Museum of Art: The Louise and Walter Arensberg Collection
• Q. I. Rahman, G. Schmeisser: Analytic theory of polynomials. Oxford University Press, 2002.
• T. Sheil-Small: Complex polynomials. Cambridge University Press, 2002.
• M. Marden: Geometry of polynomials. Second edition, American Mathematical Society, 1966.
2
This is joint research with Vania Mascioni. • On the location of critical points of polynomials. Proc. Amer. Math. Soc. 131 (2003), 253–264. • A contraction of the Lucas polygon. Proc. Amer. Math. Soc. 132 (2004), 2973–2981. • Roots and polynomials as homeomorphic spaces. Expositiones Mathematicae 24 (2006), 81–95. ◦ Results of this talk are from a paper accepted in Constructive Approximation. 3
We study polynomials with complex coefficients aj : p(z) = a0 + a1 z + . . . + an z n
We study polynomials with complex coefficients aj : p(z) = a0 + a1 z + . . . + an z n
Pn =
n
all polynomials of degree ≤ n
o
4
p ∈ Pn
n
o
Z(p) = w ∈ C : p(w) = 0
p ∈ Pn
n
o
Z(p) = w ∈ C : p(w) = 0
n
L(Pn) = all linear operators on Pn
o
p ∈ Pn
n
o
Z(p) = w ∈ C : p(w) = 0
n
L(Pn) = all linear operators on Pn
o
Vladimir Tulovsky: On perturbations of roots of polynomials. J. Analyse Math. 54 (1990), 77–89. 5
Let T ∈ L(Pn). Z(p) = Z(T p) for all non-constant p ∈ Pn
if and only if
T
?
Let T ∈ L(Pn). Z(p) = Z(T p) for all non-constant p ∈ Pn
if and only if
T = αI,
α ∈ C \ {0}
6
Let T ∈ L(Pn).
Z(p) ∩ Z(T p) 6= ∅
for all non-constant p ∈ Pn
if and only if
T
?
Let T ∈ L(Pn).
Z(p) ∩ Z(T p) 6= ∅
for all non-constant p ∈ Pn
if and only if
T = αI,
α ∈ C \ {0}
7
15
10
5
0
-15
-10
-5
0
5
10
n
o
Let T ∈ L(Pn) and D(r) = z ∈ C : |z| ≤ r . ∃ CT > 0 such that
Z(p) + D(CT ) ∩ Z(T p) 6= ∅ for all non-constant p ∈ Pn if and only if
T
?
8
15
10
5
0
-15
-10
-5
0
5
10
What is a simple example of such an operator?
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
Z(Sα p) = {−α} + Z(p).
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
Z(Sα p) = {−α} + Z(p).
α 0 αn (n) (Sαp)(z) = p(z) + p (z) + · · · + p (z) 1! n!
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
Z(Sα p) = {−α} + Z(p).
α 0 αn (n) (Sαp)(z) = p(z) + p (z) + · · · + p (z) 1! n! α αn n Sα = I + D + · · · + D 1! n!
9
n
o
Let T ∈ L(Pn) and D(r) = z ∈ C : |z| ≤ r . ∃ CT > 0 such that
Z(p) + D(CT ) ∩ Z(T p) 6= ∅ for all non-constant p ∈ Pn if and only if
n
o
Let T ∈ L(Pn) and D(r) = z ∈ C : |z| ≤ r . ∃ CT > 0 such that
Z(p) + D(CT ) ∩ Z(T p) 6= ∅ for all non-constant p ∈ Pn if and only if
T = α0 I + α1 D + · · · + αn D n ,
α0 6= 0
10
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
is true.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n ,
is true.
α0 6= 0.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n , Let φn(z) = z n.
is true.
α0 6= 0.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n , Let φn(z) = z n. n
o
Let KT = max |u| : u ∈ Z(T φn) .
is true.
α0 6= 0.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n ,
is true.
α0 6= 0.
Let φn(z) = z n. n
o
Let KT = max |u| : u ∈ Z(T φn) .
Then
Z(T p) ⊂ Z(p) + D(KT )
for all non-constant p ∈ Pn.
11
Let U and V be finite subsets of C. The Hausdorff distance is defined by: n
o
dH (U, V ) = min r > 0 : V ⊂ U + D(r), U ⊂ V + D(r) .
Let U and V be finite subsets of C. The Hausdorff distance is defined by: n
o
dH (U, V ) = min r > 0 : V ⊂ U + D(r), U ⊂ V + D(r) . Let T = α0 I + α1 D + · · · + αn Dn,
α0 6= 0.
Let U and V be finite subsets of C. The Hausdorff distance is defined by: n
o
dH (U, V ) = min r > 0 : V ⊂ U + D(r), U ⊂ V + D(r) . Let T = α0 I + α1 D + · · · + αn Dn,
α0 6= 0.
Then
n
dH Z(p), Z(T p) ≤ max KT , KT −1
o
for all non-constant p ∈ Pn.
12
Let m ∈ N. Let Πm be the set of all permutations of {1, . . . , m}.
Let m ∈ N. Let Πm be the set of all permutations of {1, . . . , m}. Let U = {u1 , . . . , um } and V = {v1 , . . . , vm } be multisets of m complex numbers.
Let m ∈ N. Let Πm be the set of all permutations of {1, . . . , m}. Let U = {u1 , . . . , um } and V = {v1 , . . . , vm } be multisets of m complex numbers.
The Fr´ echet distance is defined by: dF (U, V ) := min
σ∈Πm
max uk − vσ(k) . 1≤k≤m
13
15
10
5
0
-15
-10
-5
0
5
10
Let
T = α0 I + α1 D + · · · + αn D n ,
α0 6= 0.
Let γ1, . . . , γn be the roots of α0 z n + α1 z n−1 + · · · + αn−1 z + αn counted according to their multiplicities.
Let
T = α0 I + α1 D + · · · + αn D n ,
α0 6= 0.
Let γ1, . . . , γn be the roots of α0 z n + α1 z n−1 + · · · + αn−1 z + αn counted according to their multiplicities. Then
2 dF Z(p), Z(T p) ≤ n |γ1| + · · · + |γn|
for all non-constant p ∈ Pn.
14
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15
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
o
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
1 X c(p) = w n w∈Z(p)
o
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
1 X c(p) = w n w∈Z(p) n
ρ(p) = max |w − c(p)| : w ∈ Z(p)
o
o
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
o
1 X c(p) = w n w∈Z(p) n
ρ(p) = max |w − c(p)| : w ∈ Z(p)
spr(p) = τ (p)
τ (p) ρ(p)
o
!n−2
15
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
Z(p) ⊂ {w ∈ C : |w| = r},
Z(p0) = {0}
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
Z(p) ⊂ {w ∈ C : |w| = r},
Z(p0) = {0}
τ (p) = r,
c(p) = 0,
ρ(p) = r
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
Z(p) ⊂ {w ∈ C : |w| = r},
Z(p0) = {0}
τ (p) = r,
c(p) = 0,
spr(p) = r
n−2 r
r
ρ(p) = r
=r
16
A generalization:
Let t > 0.
Define Ht ∈ L(Pn) by (Ht p)(z) = p(z/t),
p ∈ Pn .
A generalization:
Let t > 0.
Define Ht ∈ L(Pn) by (Ht p)(z) = p(z/t),
p ∈ Pn .
Then spr(Ht p) = t spr(p).
17
αn n α2 2 D + ··· + D . Recall Sα = I + αD + 2! n!
αn n α2 2 D + ··· + D . Recall Sα = I + αD + 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
αn n α2 2 D + ··· + D . Recall Sα = I + αD + 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
Then there exists a constant ΓT > 0 such that dF
ΓT Z(Sα p), Z(T p) ≤ spr(p)
for all p ∈ Pn with n distinct roots.
18
α2 2 αn n Recall Sα = I + αD + D + ··· + D . 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
α2 2 αn n Recall Sα = I + αD + D + ··· + D . 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
A corollary:
α2 2 αn n Recall Sα = I + αD + D + ··· + D . 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
A corollary: For an arbitrary p ∈ Pn with n distinct roots:
lim dF Z(Sα Ht p), Z(T Ht p) = 0
t→+∞
19
1
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 1866 - 1944
Philadelphia Museum of Art: The Louise and Walter Arensberg Collection
• Q. I. Rahman, G. Schmeisser: Analytic theory of polynomials. Oxford University Press, 2002.
• T. Sheil-Small: Complex polynomials. Cambridge University Press, 2002.
• M. Marden: Geometry of polynomials. Second edition, American Mathematical Society, 1966.
2
This is joint research with Vania Mascioni. • On the location of critical points of polynomials. Proc. Amer. Math. Soc. 131 (2003), 253–264. • A contraction of the Lucas polygon. Proc. Amer. Math. Soc. 132 (2004), 2973–2981. • Roots and polynomials as homeomorphic spaces. Expositiones Mathematicae 24 (2006), 81–95. ◦ Results of this talk are from a paper accepted in Constructive Approximation. 3
We study polynomials with complex coefficients aj : p(z) = a0 + a1 z + . . . + an z n
We study polynomials with complex coefficients aj : p(z) = a0 + a1 z + . . . + an z n
Pn =
n
all polynomials of degree ≤ n
o
4
p ∈ Pn
n
o
Z(p) = w ∈ C : p(w) = 0
p ∈ Pn
n
o
Z(p) = w ∈ C : p(w) = 0
n
L(Pn) = all linear operators on Pn
o
p ∈ Pn
n
o
Z(p) = w ∈ C : p(w) = 0
n
L(Pn) = all linear operators on Pn
o
Vladimir Tulovsky: On perturbations of roots of polynomials. J. Analyse Math. 54 (1990), 77–89. 5
Let T ∈ L(Pn). Z(p) = Z(T p) for all non-constant p ∈ Pn
if and only if
T
?
Let T ∈ L(Pn). Z(p) = Z(T p) for all non-constant p ∈ Pn
if and only if
T = αI,
α ∈ C \ {0}
6
Let T ∈ L(Pn).
Z(p) ∩ Z(T p) 6= ∅
for all non-constant p ∈ Pn
if and only if
T
?
Let T ∈ L(Pn).
Z(p) ∩ Z(T p) 6= ∅
for all non-constant p ∈ Pn
if and only if
T = αI,
α ∈ C \ {0}
7
15
10
5
0
-15
-10
-5
0
5
10
n
o
Let T ∈ L(Pn) and D(r) = z ∈ C : |z| ≤ r . ∃ CT > 0 such that
Z(p) + D(CT ) ∩ Z(T p) 6= ∅ for all non-constant p ∈ Pn if and only if
T
?
8
15
10
5
0
-15
-10
-5
0
5
10
What is a simple example of such an operator?
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
Z(Sα p) = {−α} + Z(p).
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
Z(Sα p) = {−α} + Z(p).
α 0 αn (n) (Sαp)(z) = p(z) + p (z) + · · · + p (z) 1! n!
What is a simple example of such an operator?
(Sαp)(z) = p(α + z)
Z(Sα p) = {−α} + Z(p).
α 0 αn (n) (Sαp)(z) = p(z) + p (z) + · · · + p (z) 1! n! α αn n Sα = I + D + · · · + D 1! n!
9
n
o
Let T ∈ L(Pn) and D(r) = z ∈ C : |z| ≤ r . ∃ CT > 0 such that
Z(p) + D(CT ) ∩ Z(T p) 6= ∅ for all non-constant p ∈ Pn if and only if
n
o
Let T ∈ L(Pn) and D(r) = z ∈ C : |z| ≤ r . ∃ CT > 0 such that
Z(p) + D(CT ) ∩ Z(T p) 6= ∅ for all non-constant p ∈ Pn if and only if
T = α0 I + α1 D + · · · + αn D n ,
α0 6= 0
10
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
is true.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n ,
is true.
α0 6= 0.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n , Let φn(z) = z n.
is true.
α0 6= 0.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n , Let φn(z) = z n. n
o
Let KT = max |u| : u ∈ Z(T φn) .
is true.
α0 6= 0.
More than
Z(p) + D(CT ) ∩ Z(T p) 6= ∅
T = α0 I + α1 D + · · · + αn D n ,
is true.
α0 6= 0.
Let φn(z) = z n. n
o
Let KT = max |u| : u ∈ Z(T φn) .
Then
Z(T p) ⊂ Z(p) + D(KT )
for all non-constant p ∈ Pn.
11
Let U and V be finite subsets of C. The Hausdorff distance is defined by: n
o
dH (U, V ) = min r > 0 : V ⊂ U + D(r), U ⊂ V + D(r) .
Let U and V be finite subsets of C. The Hausdorff distance is defined by: n
o
dH (U, V ) = min r > 0 : V ⊂ U + D(r), U ⊂ V + D(r) . Let T = α0 I + α1 D + · · · + αn Dn,
α0 6= 0.
Let U and V be finite subsets of C. The Hausdorff distance is defined by: n
o
dH (U, V ) = min r > 0 : V ⊂ U + D(r), U ⊂ V + D(r) . Let T = α0 I + α1 D + · · · + αn Dn,
α0 6= 0.
Then
n
dH Z(p), Z(T p) ≤ max KT , KT −1
o
for all non-constant p ∈ Pn.
12
Let m ∈ N. Let Πm be the set of all permutations of {1, . . . , m}.
Let m ∈ N. Let Πm be the set of all permutations of {1, . . . , m}. Let U = {u1 , . . . , um } and V = {v1 , . . . , vm } be multisets of m complex numbers.
Let m ∈ N. Let Πm be the set of all permutations of {1, . . . , m}. Let U = {u1 , . . . , um } and V = {v1 , . . . , vm } be multisets of m complex numbers.
The Fr´ echet distance is defined by: dF (U, V ) := min
σ∈Πm
max uk − vσ(k) . 1≤k≤m
13
15
10
5
0
-15
-10
-5
0
5
10
Let
T = α0 I + α1 D + · · · + αn D n ,
α0 6= 0.
Let γ1, . . . , γn be the roots of α0 z n + α1 z n−1 + · · · + αn−1 z + αn counted according to their multiplicities.
Let
T = α0 I + α1 D + · · · + αn D n ,
α0 6= 0.
Let γ1, . . . , γn be the roots of α0 z n + α1 z n−1 + · · · + αn−1 z + αn counted according to their multiplicities. Then
2 dF Z(p), Z(T p) ≤ n |γ1| + · · · + |γn|
for all non-constant p ∈ Pn.
14
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15
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
o
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
1 X c(p) = w n w∈Z(p)
o
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
1 X c(p) = w n w∈Z(p) n
ρ(p) = max |w − c(p)| : w ∈ Z(p)
o
o
Let p ∈ Pn be a polynomial with n distinct roots. Then Z(p) ∩ Z(p0) = ∅. Define τ (p) = min |w − v| : w ∈ Z(p), v ∈ Z(p0) n
o
1 X c(p) = w n w∈Z(p) n
ρ(p) = max |w − c(p)| : w ∈ Z(p)
spr(p) = τ (p)
τ (p) ρ(p)
o
!n−2
15
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
Z(p) ⊂ {w ∈ C : |w| = r},
Z(p0) = {0}
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
Z(p) ⊂ {w ∈ C : |w| = r},
Z(p0) = {0}
τ (p) = r,
c(p) = 0,
ρ(p) = r
A trivial example: Let r > 0. p(z) = z n − rn,
p0(z) = nz n−1
Z(p) ⊂ {w ∈ C : |w| = r},
Z(p0) = {0}
τ (p) = r,
c(p) = 0,
spr(p) = r
n−2 r
r
ρ(p) = r
=r
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A generalization:
Let t > 0.
Define Ht ∈ L(Pn) by (Ht p)(z) = p(z/t),
p ∈ Pn .
A generalization:
Let t > 0.
Define Ht ∈ L(Pn) by (Ht p)(z) = p(z/t),
p ∈ Pn .
Then spr(Ht p) = t spr(p).
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αn n α2 2 D + ··· + D . Recall Sα = I + αD + 2! n!
αn n α2 2 D + ··· + D . Recall Sα = I + αD + 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
αn n α2 2 D + ··· + D . Recall Sα = I + αD + 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
Then there exists a constant ΓT > 0 such that dF
ΓT Z(Sα p), Z(T p) ≤ spr(p)
for all p ∈ Pn with n distinct roots.
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α2 2 αn n Recall Sα = I + αD + D + ··· + D . 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
α2 2 αn n Recall Sα = I + αD + D + ··· + D . 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
A corollary:
α2 2 αn n Recall Sα = I + αD + D + ··· + D . 2! n! Let
T = I + αD + α2 D2 + · · · + αn Dn.
A corollary: For an arbitrary p ∈ Pn with n distinct roots:
lim dF Z(Sα Ht p), Z(T Ht p) = 0
t→+∞
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