Erdős on polynomials

Transcription

1 Erdős on polynomials Vilmos Totik University of Szeged and University of South Florida Vilmos Totik (SZTE and USF) Polynomials 1 / *

2 Erdős on polynomials Vilmos Totik (SZTE and USF) Polynomials 2 / *

3 Erdős on polynomials Interpolation Vilmos Totik (SZTE and USF) Polynomials 2 / *

4 Erdős on polynomials Interpolation Discrepancy theorems for zeros Vilmos Totik (SZTE and USF) Polynomials 2 / *

5 Erdős on polynomials Interpolation Discrepancy theorems for zeros Inequalities Vilmos Totik (SZTE and USF) Polynomials 2 / *

6 Erdős on polynomials Interpolation Discrepancy theorems for zeros Inequalities Growth of polynomials Vilmos Totik (SZTE and USF) Polynomials 2 / *

7 Erdős on polynomials Interpolation Discrepancy theorems for zeros Inequalities Growth of polynomials Geometric problems for lemniscates Vilmos Totik (SZTE and USF) Polynomials 2 / *

8 Erdős on polynomials Interpolation Discrepancy theorems for zeros Inequalities Growth of polynomials Geometric problems for lemniscates Orthogonal polynomials Vilmos Totik (SZTE and USF) Polynomials 2 / *

9 Erdős on polynomials Interpolation Discrepancy theorems for zeros Inequalities Growth of polynomials Geometric problems for lemniscates Orthogonal polynomials Spacing of zeros Vilmos Totik (SZTE and USF) Polynomials 2 / *

10 Erdős on polynomials Interpolation Discrepancy theorems for zeros Inequalities Growth of polynomials Geometric problems for lemniscates Orthogonal polynomials Spacing of zeros Geometry of zeros of derivatives Vilmos Totik (SZTE and USF) Polynomials 2 / *

11 Erdős on polynomials Interpolation Discrepancy theorems for zeros Inequalities Growth of polynomials Geometric problems for lemniscates Orthogonal polynomials Spacing of zeros Geometry of zeros of derivatives Polynomials with integer coefficients Vilmos Totik (SZTE and USF) Polynomials 2 / *

12 Erdős on polynomials Interpolation (Lubinsky s and Vértesi s surveys) Discrepancy theorems for zeros Inequalities (Erdélyi s survey) Growth of polynomials Geometric problems for lemniscates (Eremenko-Hayman s and Borwein s surveys) Orthogonal polynomials Spacing of zeros Geometry of zeros of derivatives Polynomials with integer coefficients (Borwein s and Erdélyi s surveys) Vilmos Totik (SZTE and USF) Polynomials 3 / *

13 Chebyshev polynomials Vilmos Totik (SZTE and USF) Polynomials 4 / *

14 Chebyshev polynomials P. L. Chebyshev: replace x 4 in [ 1,1] by combination of smaller powers Vilmos Totik (SZTE and USF) Polynomials 4 / *

15 Chebyshev polynomials P. L. Chebyshev: replace x 4 in [ 1,1] by combination of smaller powers Approximate x n by linear combination of smaller powers Vilmos Totik (SZTE and USF) Polynomials 4 / *

16 Chebyshev polynomials P. L. Chebyshev: replace x 4 in [ 1,1] by combination of smaller powers Approximate x n by linear combination of smaller powers t n = inf P n(x)=x n + P n [ 1,1] Vilmos Totik (SZTE and USF) Polynomials 4 / *

17 Chebyshev polynomials P. L. Chebyshev: replace x 4 in [ 1,1] by combination of smaller powers Approximate x n by linear combination of smaller powers t n = inf P n(x)=x n + P n [ 1,1] P n K = sup P n (z) z K Vilmos Totik (SZTE and USF) Polynomials 4 / *

18 Chebyshev polynomials P. L. Chebyshev: replace x 4 in [ 1,1] by combination of smaller powers Approximate x n by linear combination of smaller powers t n = inf P n(x)=x n + P n [ 1,1] P n K = sup P n (z) z K t n = 2 2 n Vilmos Totik (SZTE and USF) Polynomials 4 / *

19 Chebyshev polynomials P. L. Chebyshev: replace x 4 in [ 1,1] by combination of smaller powers Approximate x n by linear combination of smaller powers t n = inf P n(x)=x n + P n [ 1,1] P n K = sup P n (z) z K t n = 2 2 n T n (z) = 1 cos(narccosx) 2n 1 Vilmos Totik (SZTE and USF) Polynomials 4 / *

20 Zeros of the Chebyshev polynomials Uniform spacing of zeros -1 1 Vilmos Totik (SZTE and USF) Polynomials 5 / *

21 The Erdős-Turán discrepancy theorem Vilmos Totik (SZTE and USF) Polynomials 6 / *

22 The Erdős-Turán discrepancy theorem Suppose P n (x) = x n + has all its zeros in [ 1,1] Vilmos Totik (SZTE and USF) Polynomials 6 / *

23 The Erdős-Turán discrepancy theorem Suppose P n (x) = x n + has all its zeros in [ 1,1] Let x j be the zeros Vilmos Totik (SZTE and USF) Polynomials 6 / *

24 The Erdős-Turán discrepancy theorem Suppose P n (x) = x n + has all its zeros in [ 1,1] Let x j be the zeros P n [ 1,1] A n 2 n Vilmos Totik (SZTE and USF) Polynomials 6 / *

25 The Erdős-Turán discrepancy theorem Suppose P n (x) = x n + has all its zeros in [ 1,1] Let x j be the zeros P n [ 1,1] A n 2 n Th. (Erdős-Turán, 1940) For any 1 a < b 1 #{x j (a,b)} n arcsinb arcsina π 8 logan n Vilmos Totik (SZTE and USF) Polynomials 6 / *

26 Equilibrium measures Vilmos Totik (SZTE and USF) Polynomials 7 / *

27 Equilibrium measures E C compact Vilmos Totik (SZTE and USF) Polynomials 7 / *

28 Equilibrium measures E C compact µ unit Borel-measure on E, Vilmos Totik (SZTE and USF) Polynomials 7 / *

29 Equilibrium measures E C compact µ unit Borel-measure on E, its logarithmic energy is 1 I(µ) = log z t dµ(z)dµ(t) Vilmos Totik (SZTE and USF) Polynomials 7 / *

30 Equilibrium measures E C compact µ unit Borel-measure on E, its logarithmic energy is 1 I(µ) = log z t dµ(z)dµ(t) If this is finite for some µ, then there is a unique minimizing measure µ E, called equilibrium measure of E Vilmos Totik (SZTE and USF) Polynomials 7 / *

31 Equilibrium measures E C compact µ unit Borel-measure on E, its logarithmic energy is 1 I(µ) = log z t dµ(z)dµ(t) If this is finite for some µ, then there is a unique minimizing measure µ E, called equilibrium measure of E Examples: dµ [ 1,1] (t) = 1 π 1 t 2dt Vilmos Totik (SZTE and USF) Polynomials 7 / *

32 Equilibrium measures E C compact µ unit Borel-measure on E, its logarithmic energy is 1 I(µ) = log z t dµ(z)dµ(t) If this is finite for some µ, then there is a unique minimizing measure µ E, called equilibrium measure of E Examples: If C 1 is the unit circle, then dµ [ 1,1] (t) = 1 π 1 t 2dt dµ C1 (e it ) = 1 2π dt is the normalized arc measure Vilmos Totik (SZTE and USF) Polynomials 7 / *

33 Logarithmic capacity With the minimal energy I(K) = inf µ I(µ) Vilmos Totik (SZTE and USF) Polynomials 8 / *

34 Logarithmic capacity With the minimal energy I(K) = inf µ I(µ) define the logarithmic capacity as cap(k) = e I(K) Vilmos Totik (SZTE and USF) Polynomials 8 / *

35 Logarithmic capacity With the minimal energy I(K) = inf µ I(µ) define the logarithmic capacity as cap(k) = e I(K) Examples: If K is a disk/circle of radius r then c(k) = r Vilmos Totik (SZTE and USF) Polynomials 8 / *

36 Logarithmic capacity With the minimal energy I(K) = inf µ I(µ) define the logarithmic capacity as cap(k) = e I(K) Examples: If K is a disk/circle of radius r then c(k) = r cap([ 1,1]) = 1/2 Vilmos Totik (SZTE and USF) Polynomials 8 / *

37 Logarithmic capacity With the minimal energy I(K) = inf µ I(µ) define the logarithmic capacity as cap(k) = e I(K) Examples: If K is a disk/circle of radius r then c(k) = r cap([ 1,1]) = 1/2 If P n (z) = z n +, then P n K cap(k) n Vilmos Totik (SZTE and USF) Polynomials 8 / *

38 Transfinite diameter, Chebyshev constants K C compact δ n = sup z 1,...,z n K z i z j i j Vilmos Totik (SZTE and USF) Polynomials 9 / *

39 Transfinite diameter, Chebyshev constants K C compact δ n = sup z 1,...,z n K z i z j i j Transfinite diameter: δ 1/n(n 1) n δ(k) Vilmos Totik (SZTE and USF) Polynomials 9 / *

40 Transfinite diameter, Chebyshev constants K C compact δ n = sup z 1,...,z n K z i z j i j Transfinite diameter: δ 1/n(n 1) n δ(k) t n = inf z n + K Vilmos Totik (SZTE and USF) Polynomials 9 / *

41 Transfinite diameter, Chebyshev constants K C compact δ n = sup z 1,...,z n K z i z j i j Transfinite diameter: δ 1/n(n 1) n δ(k) t n = inf z n + K Chebyshev constant t 1/n n t(k) Vilmos Totik (SZTE and USF) Polynomials 9 / *

42 Transfinite diameter, Chebyshev constants K C compact δ n = sup z 1,...,z n K z i z j i j Transfinite diameter: δ 1/n(n 1) n δ(k) t n = inf z n + K Chebyshev constant t 1/n n t(k) Fekete, Zygmund, Szegő: cap(k) = δ(k) = t(k) Vilmos Totik (SZTE and USF) Polynomials 9 / *

43 The Erdős-Turán discrepancy theorem Vilmos Totik (SZTE and USF) Polynomials 10 / *

44 The Erdős-Turán discrepancy theorem P n (x) = x n +, Vilmos Totik (SZTE and USF) Polynomials 10 / *

45 The Erdős-Turán discrepancy theorem P n (x) = x n +, P n [ 1,1] A n /2 n Vilmos Totik (SZTE and USF) Polynomials 10 / *

46 The Erdős-Turán discrepancy theorem P n (x) = x n +, P n [ 1,1] A n /2 n The Erdős-Turán theorem: For any 1 a < b 1 #{x j (a,b)} b 1 n a π 1 x 2dx 8 logan n (1) Vilmos Totik (SZTE and USF) Polynomials 10 / *

47 The Erdős-Turán discrepancy theorem P n (x) = x n +, P n [ 1,1] A n /2 n The Erdős-Turán theorem: For any 1 a < b 1 #{x j (a,b)} b 1 n a π 1 x 2dx 8 logan n Normalized zero distribution ν n = 1 n j δ x j (1) Vilmos Totik (SZTE and USF) Polynomials 10 / *

48 The Erdős-Turán discrepancy theorem P n (x) = x n +, P n [ 1,1] A n /2 n The Erdős-Turán theorem: For any 1 a < b 1 #{x j (a,b)} b 1 n a π 1 x 2dx 8 logan n Normalized zero distribution ν n = 1 n j δ x j Equivalent form of (1): with the Chebyshev distribution dµ [ 1,1] (x) = 1 π 1 x 2dx (1) Vilmos Totik (SZTE and USF) Polynomials 10 / *

49 The Erdős-Turán discrepancy theorem P n (x) = x n +, P n [ 1,1] A n /2 n The Erdős-Turán theorem: For any 1 a < b 1 #{x j (a,b)} b 1 n a π 1 x 2dx 8 logan n Normalized zero distribution ν n = 1 n j δ x j Equivalent form of (1): with the Chebyshev distribution for any I [ 1,1] dµ [ 1,1] (x) = 1 π 1 x 2dx (1) νn (I) µ [ 1,1] (I) 8 logan n Vilmos Totik (SZTE and USF) Polynomials 10 / *

50 The Erdős-Turán discrepancy theorem P n (x) = x n +, P n [ 1,1] A n /2 n The Erdős-Turán theorem: For any 1 a < b 1 #{x j (a,b)} b 1 n a π 1 x 2dx 8 logan n Normalized zero distribution ν n = 1 n j δ x j Equivalent form of (1): with the Chebyshev distribution for any I [ 1,1] dµ [ 1,1] (x) = 1 π 1 x 2dx (1) νn (I) µ [ 1,1] (I) 8 logan n Vilmos Totik (SZTE and USF) Polynomials 10 / *

51 A general discrepancy theorem Let K be a finite union of smooth Jordan arcs Vilmos Totik (SZTE and USF) Polynomials 11 / *

52 A general discrepancy theorem Let K be a finite union of smooth Jordan arcs J Vilmos Totik (SZTE and USF) Polynomials 11 / *

53 A general discrepancy theorem Let K be a finite union of smooth Jordan arcs J Vilmos Totik (SZTE and USF) Polynomials 11 / *

54 A general discrepancy theorem Let K be a finite union of smooth Jordan arcs J J* Vilmos Totik (SZTE and USF) Polynomials 12 / *

55 A general discrepancy theorem Let K be a finite union of smooth Jordan arcs J J* Recall: if P n (z) = z n +, then P n K cap(k) n Vilmos Totik (SZTE and USF) Polynomials 12 / *

56 A general discrepancy theorem K finite union of smooth Jordan arcs Vilmos Totik (SZTE and USF) Polynomials 13 / *

57 A general discrepancy theorem K finite union of smooth Jordan arcs P n (z) = z n + Vilmos Totik (SZTE and USF) Polynomials 13 / *

58 A general discrepancy theorem K finite union of smooth Jordan arcs P n (z) = z n + P n K A n cap(k) n Vilmos Totik (SZTE and USF) Polynomials 13 / *

59 A general discrepancy theorem K finite union of smooth Jordan arcs P n (z) = z n + P n K A n cap(k) n ν n normalized zero distribution Vilmos Totik (SZTE and USF) Polynomials 13 / *

60 A general discrepancy theorem K finite union of smooth Jordan arcs P n (z) = z n + P n K A n cap(k) n ν n normalized zero distribution Th. (Andrievskii-Blatt, ) For any J K ν n (J ) µ K (J logan ) C. n Vilmos Totik (SZTE and USF) Polynomials 13 / *

61 A general discrepancy theorem K finite union of smooth Jordan arcs P n (z) = z n + P n K A n cap(k) n ν n normalized zero distribution Th. (Andrievskii-Blatt, ) For any J K ν n (J ) µ K (J logan ) C. n In particular, if P n 1/n K cap(k), then ν n µ K Vilmos Totik (SZTE and USF) Polynomials 13 / *

62 The Jentzsch-Szegő theorem Vilmos Totik (SZTE and USF) Polynomials 14 / *

63 The Jentzsch-Szegő theorem No such results are true for Jordan curves: Vilmos Totik (SZTE and USF) Polynomials 14 / *

64 The Jentzsch-Szegő theorem No such results are true for Jordan curves: z n on the unit circle Vilmos Totik (SZTE and USF) Polynomials 14 / *

65 The Jentzsch-Szegő theorem No such results are true for Jordan curves: z n on the unit circle Jentsch, 1918: If the radius of convergence of j=0 a jz j is 1, then the zeros of the partial sums n 0 a jz j, n = 1,2,... are dense at the unit circle Vilmos Totik (SZTE and USF) Polynomials 14 / *

66 The Jentzsch-Szegő theorem No such results are true for Jordan curves: z n on the unit circle Jentsch, 1918: If the radius of convergence of j=0 a jz j is 1, then the zeros of the partial sums n 0 a jz j, n = 1,2,... are dense at the unit circle Szegő, 1922: There is a sequence n 1 < n 2 < such that if z j,n = r j,n e iθ j,n, 1 j n are the zeros of n 0 a jz j, then {θ j,nk } n k 0 is asymptotically uniformly distributed (and r j,nk 1 for most j) Vilmos Totik (SZTE and USF) Polynomials 14 / *

67 The second discrepancy theorem P n (z) = a n z n + +a 0, C 1 = { z = 1} Vilmos Totik (SZTE and USF) Polynomials 15 / *

68 The second discrepancy theorem P n (z) = a n z n + +a 0, C 1 = { z = 1} z j = r j e iθ j are the zeros Vilmos Totik (SZTE and USF) Polynomials 15 / *

69 The second discrepancy theorem P n (z) = a n z n + +a 0, C 1 = { z = 1} z j = r j e iθ j are the zeros Th. (Erdős-Turán, 1950) For any J [0,2π] Vilmos Totik (SZTE and USF) Polynomials 15 / *

70 The second discrepancy theorem P n (z) = a n z n + +a 0, C 1 = { z = 1} z j = r j e iθ j are the zeros Th. (Erdős-Turán, 1950) For any J [0,2π] #{θ j J} J n 2π 16 log P n C1 / a 0 a n n Vilmos Totik (SZTE and USF) Polynomials 15 / *

71 The second discrepancy theorem P n (z) = a n z n + +a 0, C 1 = { z = 1} z j = r j e iθ j are the zeros Th. (Erdős-Turán, 1950) For any J [0,2π] #{θ j J} J n 2π 16 log P n C1 / a 0 a n n Note P n C1 j a j Vilmos Totik (SZTE and USF) Polynomials 15 / *

72 Number of real zeros of a polynomial Vilmos Totik (SZTE and USF) Polynomials 16 / *

73 Number of real zeros of a polynomial #{θ j J} n J log( 2π 16 j a j / ) a 0 a n, n Vilmos Totik (SZTE and USF) Polynomials 16 / *

74 Number of real zeros of a polynomial #{θ j J} n J log( 2π 16 j a j / ) a 0 a n, n Consequence: there are at most ( ) 32 nlog a j / a0 a n real zeros of P n j Vilmos Totik (SZTE and USF) Polynomials 16 / *

75 Number of real zeros of a polynomial #{θ j J} n J log( 2π 16 j a j / ) a 0 a n, n Consequence: there are at most ( ) 32 nlog a j / a0 a n real zeros of P n Better than previous estimates of Bloch, Pólya, Schmidt, basically a theorem of Schur, Szegő j Vilmos Totik (SZTE and USF) Polynomials 16 / *

76 The Szegő theorem Vilmos Totik (SZTE and USF) Polynomials 17 / *

77 The Szegő theorem j=0 a jz j, a 0 0 Vilmos Totik (SZTE and USF) Polynomials 17 / *

78 The Szegő theorem j=0 a jz j, a 0 0 Radius of convergence 1 limsup n a n 1/n = 1 Vilmos Totik (SZTE and USF) Polynomials 17 / *

79 The Szegő theorem j=0 a jz j, a 0 0 Radius of convergence 1 limsup n a n 1/n = 1 There is a subsequence {n k } with C nk := ( nk j=0 a j a0 a n ) 1/nk 1 Vilmos Totik (SZTE and USF) Polynomials 17 / *

80 The Szegő theorem j=0 a jz j, a 0 0 Radius of convergence 1 limsup n a n 1/n = 1 There is a subsequence {n k } with C nk := ( nk j=0 a j a0 a n ) 1/nk 1 z j,n = r j,n e iθ j,n are the zeros of n j=0 a jz j Vilmos Totik (SZTE and USF) Polynomials 17 / *

81 The Szegő theorem j=0 a jz j, a 0 0 Radius of convergence 1 limsup n a n 1/n = 1 There is a subsequence {n k } with C nk := ( nk j=0 a j a0 a n ) 1/nk 1 z j,n = r j,n e iθ j,n are the zeros of n j=0 a jz j #{θ j,nk J} n k J 2π 16 logc nk 0 Vilmos Totik (SZTE and USF) Polynomials 17 / *

82 General discrepancy theorems Vilmos Totik (SZTE and USF) Polynomials 18 / *

83 General discrepancy theorems J J* z 0 Vilmos Totik (SZTE and USF) Polynomials 18 / *

84 General discrepancy theorems J J* z 0 Andrievskii-Blatt, : If Γ is a smooth Jordan curve, z 0 a fixed point inside, P n (z) = z n + +a n, and Vilmos Totik (SZTE and USF) Polynomials 18 / *

85 General discrepancy theorems J J* z 0 Andrievskii-Blatt, : If Γ is a smooth Jordan curve, z 0 a fixed point inside, P n (z) = z n + +a n, and B n = P n Γ / cap(γ) n P n (z 0 ), then for all J Γ Vilmos Totik (SZTE and USF) Polynomials 18 / *

86 General discrepancy theorems J J* z 0 Andrievskii-Blatt, : If Γ is a smooth Jordan curve, z 0 a fixed point inside, P n (z) = z n + +a n, and B n = P n Γ / cap(γ) n P n (z 0 ), then for all J Γ #{z j J } µ Γ (J) n C logbn 0 n Vilmos Totik (SZTE and USF) Polynomials 18 / *

87 It is true for a family of Jordan curves Vilmos Totik (SZTE and USF) Polynomials 18 / * General discrepancy theorems J J* z 0 Andrievskii-Blatt, : If Γ is a smooth Jordan curve, z 0 a fixed point inside, P n (z) = z n + +a n, and B n = P n Γ / cap(γ) n P n (z 0 ), then for all J Γ #{z j J } µ Γ (J) n C logbn 0 n

88 Orthogonal polynomials ρ a measure with compact support Vilmos Totik (SZTE and USF) Polynomials 19 / *

89 Orthogonal polynomials ρ a measure with compact support p n (z) = γ n z n + orthonormal polynomials with respect to ρ: { 0 if n m p n p m dρ = 1 if n = m Vilmos Totik (SZTE and USF) Polynomials 19 / *

90 Orthogonal polynomials ρ a measure with compact support p n (z) = γ n z n + orthonormal polynomials with respect to ρ: { 0 if n m p n p m dρ = 1 if n = m Erdős-Turán were among the first to get results for general weights Vilmos Totik (SZTE and USF) Polynomials 19 / *

91 Orthogonal polynomials ρ a measure with compact support p n (z) = γ n z n + orthonormal polynomials with respect to ρ: { 0 if n m p n p m dρ = 1 if n = m Erdős-Turán were among the first to get results for general weights Th. (Erdős-Turán, 1940) If the support of ρ is [ 1,1], dρ(x) = w(x)dx and w > 0 almost everywhere, Vilmos Totik (SZTE and USF) Polynomials 19 / *

92 Orthogonal polynomials ρ a measure with compact support p n (z) = γ n z n + orthonormal polynomials with respect to ρ: { 0 if n m p n p m dρ = 1 if n = m Erdős-Turán were among the first to get results for general weights Th. (Erdős-Turán, 1940) If the support of ρ is [ 1,1], dρ(x) = w(x)dx and w > 0 almost everywhere, then the asymptotic zero distribution of p n is the Chebyshev distribution Vilmos Totik (SZTE and USF) Polynomials 19 / *

93 Orthogonal polynomials ρ a measure with compact support p n (z) = γ n z n + orthonormal polynomials with respect to ρ: { 0 if n m p n p m dρ = 1 if n = m Erdős-Turán were among the first to get results for general weights Th. (Erdős-Turán, 1940) If the support of ρ is [ 1,1], dρ(x) = w(x)dx and w > 0 almost everywhere, then the asymptotic zero distribution of p n is the Chebyshev distribution p n (z) 1/n z + z 2 1, z [ 1,1] Vilmos Totik (SZTE and USF) Polynomials 19 / *

94 Orthogonal polynomials w > 0 almost everywhere on [ 1, 1] implies classical behavior Vilmos Totik (SZTE and USF) Polynomials 20 / *

95 Orthogonal polynomials w > 0 almost everywhere on [ 1, 1] implies classical behavior Rakhmanov, 1982: actually p n+1 (z) p n (z) z + z 2 1 Vilmos Totik (SZTE and USF) Polynomials 20 / *

96 Orthogonal polynomials w > 0 almost everywhere on [ 1, 1] implies classical behavior Rakhmanov, 1982: actually p n+1 (z) p n (z) z + z 2 1 Widom, 1967: If the support is not connected, then such external behavior is not possible Vilmos Totik (SZTE and USF) Polynomials 20 / *

97 Orthogonal polynomials w > 0 almost everywhere on [ 1, 1] implies classical behavior Rakhmanov, 1982: actually p n+1 (z) p n (z) z + z 2 1 Widom, 1967: If the support is not connected, then such external behavior is not possible Assume S = supp(ρ) has connected complement and empty interior (e.g. S R) Vilmos Totik (SZTE and USF) Polynomials 20 / *

98 Orthogonal polynomials w > 0 almost everywhere on [ 1, 1] implies classical behavior Rakhmanov, 1982: actually p n+1 (z) p n (z) z + z 2 1 Widom, 1967: If the support is not connected, then such external behavior is not possible Assume S = supp(ρ) has connected complement and empty interior (e.g. S R) Always 1 cap(s) liminfγ1/n n Vilmos Totik (SZTE and USF) Polynomials 20 / *

99 Regular behavior Vilmos Totik (SZTE and USF) Polynomials 21 / *

100 Regular behavior The following are equivalent (Stahl-T., 1991, Ullman, Erdős, Widom,...): Vilmos Totik (SZTE and USF) Polynomials 21 / *

101 Regular behavior The following are equivalent (Stahl-T., 1991, Ullman, Erdős, Widom,...): The asymptotic zero distribution of p n is the equilibrium distribution µ S of the support S Vilmos Totik (SZTE and USF) Polynomials 21 / *

102 Regular behavior The following are equivalent (Stahl-T., 1991, Ullman, Erdős, Widom,...): The asymptotic zero distribution of p n is the equilibrium distribution µ S of the support S p n (z) 1/n G S (z) z S Vilmos Totik (SZTE and USF) Polynomials 21 / *

103 Regular behavior The following are equivalent (Stahl-T., 1991, Ullman, Erdős, Widom,...): The asymptotic zero distribution of p n is the equilibrium distribution µ S of the support S p n (z) 1/n G S (z) z S γ 1/n n 1/cap(S) Vilmos Totik (SZTE and USF) Polynomials 21 / *

104 Regular behavior The following are equivalent (Stahl-T., 1991, Ullman, Erdős, Widom,...): The asymptotic zero distribution of p n is the equilibrium distribution µ S of the support S p n (z) 1/n G S (z) z S γ 1/n n 1/cap(S) sup P n 1/n P n S P n 1/n L q (ρ) 1, q > 0 Vilmos Totik (SZTE and USF) Polynomials 21 / *

105 Regular behavior The following are equivalent (Stahl-T., 1991, Ullman, Erdős, Widom,...): The asymptotic zero distribution of p n is the equilibrium distribution µ S of the support S p n (z) 1/n G S (z) z S γ 1/n n 1/cap(S) sup P n In this case we say ρ Reg 1/n P n S P n 1/n L q (ρ) 1, q > 0 Vilmos Totik (SZTE and USF) Polynomials 21 / *

106 Regular behavior Vilmos Totik (SZTE and USF) Polynomials 22 / *

107 Regular behavior ρ Reg the measure is not too thin on any part of its support Vilmos Totik (SZTE and USF) Polynomials 22 / *

108 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically Vilmos Totik (SZTE and USF) Polynomials 22 / *

109 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Vilmos Totik (SZTE and USF) Polynomials 22 / *

110 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Erdős-Turán criterion for regularity on [ 1, 1]: dρ(z) = w(z)dz with w(z) > 0 a.e. Vilmos Totik (SZTE and USF) Polynomials 22 / *

111 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Erdős-Turán criterion for regularity on [ 1, 1]: dρ(z) = w(z)dz with w(z) > 0 a.e. General Erdős-Turán criterion for regularity: dρ(z) = w(z)dµ S (z) with w(z) > 0 µ S -a.e. Vilmos Totik (SZTE and USF) Polynomials 22 / *

112 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Erdős-Turán criterion for regularity on [ 1, 1]: dρ(z) = w(z)dz with w(z) > 0 a.e. General Erdős-Turán criterion for regularity: dρ(z) = w(z)dµ S (z) with w(z) > 0 µ S -a.e. Many weaker criteria exist Vilmos Totik (SZTE and USF) Polynomials 22 / *

113 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Erdős-Turán criterion for regularity on [ 1, 1]: dρ(z) = w(z)dz with w(z) > 0 a.e. General Erdős-Turán criterion for regularity: dρ(z) = w(z)dµ S (z) with w(z) > 0 µ S -a.e. Many weaker criteria exist No necessary and sufficient condition is known Vilmos Totik (SZTE and USF) Polynomials 22 / *

114 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Erdős-Turán criterion for regularity on [ 1, 1]: dρ(z) = w(z)dz with w(z) > 0 a.e. General Erdős-Turán criterion for regularity: dρ(z) = w(z)dµ S (z) with w(z) > 0 µ S -a.e. Many weaker criteria exist No necessary and sufficient condition is known Erdős conjectured: if S = [ 1,1] and dρ(x) = w(x)dx with a bounded w, then ρ Reg Vilmos Totik (SZTE and USF) Polynomials 22 / *

115 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Erdős-Turán criterion for regularity on [ 1, 1]: dρ(z) = w(z)dz with w(z) > 0 a.e. General Erdős-Turán criterion for regularity: dρ(z) = w(z)dµ S (z) with w(z) > 0 µ S -a.e. Many weaker criteria exist No necessary and sufficient condition is known Erdős conjectured: if S = [ 1,1] and dρ(x) = w(x)dx with a bounded w, then ρ Reg if and only if cap(e ǫ ) 1/2 as ǫ 0 where E ǫ is any set obtained from {x w(x) > 0} by removing a subset of measure < ǫ. Vilmos Totik (SZTE and USF) Polynomials 22 / *

116 Regular behavior ρ Reg the measure is not too thin on any part of its support orthogonal polynomials behave non-pathologically ρ Reg is a very weak assumption on the measure Erdős-Turán criterion for regularity on [ 1, 1]: dρ(z) = w(z)dz with w(z) > 0 a.e. General Erdős-Turán criterion for regularity: dρ(z) = w(z)dµ S (z) with w(z) > 0 µ S -a.e. Many weaker criteria exist No necessary and sufficient condition is known Erdős conjectured: if S = [ 1,1] and dρ(x) = w(x)dx with a bounded w, then ρ Reg if and only if cap(e ǫ ) 1/2 as ǫ 0 where E ǫ is any set obtained from {x w(x) > 0} by removing a subset of measure < ǫ. Sufficiency follows from some strong regularity criteria Vilmos Totik (SZTE and USF) Polynomials 22 / *

117 Spacing of zeros Vilmos Totik (SZTE and USF) Polynomials 23 / *

118 Spacing of zeros dρ(x) = w(x)dx a measure on [ 1,1], Vilmos Totik (SZTE and USF) Polynomials 23 / *

119 Spacing of zeros dρ(x) = w(x)dx a measure on [ 1,1], p n orthogonal polynomials, Vilmos Totik (SZTE and USF) Polynomials 23 / *

120 Spacing of zeros dρ(x) = w(x)dx a measure on [ 1,1], p n orthogonal polynomials, x j,n = cosθ j,n are the zeros Vilmos Totik (SZTE and USF) Polynomials 23 / *

121 Spacing of zeros dρ(x) = w(x)dx a measure on [ 1,1], p n orthogonal polynomials, x j,n = cosθ j,n are the zeros Rough spacing: θ j 1 θ j 1 n, i.e. c 1 n θ j 1 θ j c 2 n Vilmos Totik (SZTE and USF) Polynomials 23 / *

122 Spacing of zeros dρ(x) = w(x)dx a measure on [ 1,1], p n orthogonal polynomials, x j,n = cosθ j,n are the zeros Rough spacing: θ j 1 θ j 1 n, i.e. c 1 n θ j 1 θ j c 2 n Fine spacing: θ j 1 θ j π n, i.e. n(θ j 1 θ j ) π n Vilmos Totik (SZTE and USF) Polynomials 23 / *

123 Spacing of zeros dρ(x) = w(x)dx a measure on [ 1,1], p n orthogonal polynomials, x j,n = cosθ j,n are the zeros Rough spacing: θ j 1 θ j 1 n, i.e. c 1 n θ j 1 θ j c 2 n Fine spacing: θ j 1 θ j π n, i.e. n(θ j 1 θ j ) π n Classical (Jacobi) polynomials obey fine spacing inside the interval of orthogonality Vilmos Totik (SZTE and USF) Polynomials 23 / *

124 Spacing of zeros dρ(x) = w(x)dx a measure on [ 1,1], p n orthogonal polynomials, x j,n = cosθ j,n are the zeros Rough spacing: θ j 1 θ j 1 n, i.e. c 1 n θ j 1 θ j c 2 n Fine spacing: θ j 1 θ j π n, i.e. n(θ j 1 θ j ) π n Classical (Jacobi) polynomials obey fine spacing inside the interval of orthogonality Vilmos Totik (SZTE and USF) Polynomials 23 / *

125 Rough spacing of zeros Erdős and Turán had many results in s on rough spacing Vilmos Totik (SZTE and USF) Polynomials 24 / *

126 Rough spacing of zeros Erdős and Turán had many results in s on rough spacing Mastroianni-T., 2005: rough spacing ρ doubling Vilmos Totik (SZTE and USF) Polynomials 24 / *

127 Rough spacing of zeros Erdős and Turán had many results in s on rough spacing Mastroianni-T., 2005: rough spacing ρ doubling ρ is doubling if ρ(2i) Cρ(I) for I [ 1,1] Vilmos Totik (SZTE and USF) Polynomials 24 / *

128 Fine zero spacing Vilmos Totik (SZTE and USF) Polynomials 25 / *

129 Fine zero spacing Th. (Erdős-Turán, 1940) If w > 0 continuous on [ 1,1], then θ j 1 θ j π n θ j [ǫ,π ǫ] (2) Vilmos Totik (SZTE and USF) Polynomials 25 / *

130 Fine zero spacing Th. (Erdős-Turán, 1940) If w > 0 continuous on [ 1,1], then θ j 1 θ j π n θ j [ǫ,π ǫ] (2) No longer true if w can vanish! Vilmos Totik (SZTE and USF) Polynomials 25 / *

131 Fine zero spacing Th. (Erdős-Turán, 1940) If w > 0 continuous on [ 1,1], then θ j 1 θ j π n θ j [ǫ,π ǫ] (2) No longer true if w can vanish! A form of universality in random matrix theory asymptotics for the kernel n p k (z +a/n)p k (z +b/n) a,b C k=0 Vilmos Totik (SZTE and USF) Polynomials 25 / *

132 Fine zero spacing Th. (Erdős-Turán, 1940) If w > 0 continuous on [ 1,1], then θ j 1 θ j π n θ j [ǫ,π ǫ] (2) No longer true if w can vanish! A form of universality in random matrix theory asymptotics for the kernel n p k (z +a/n)p k (z +b/n) a,b C k=0 Lubinsky, 2009: universality is true under the ρ Reg global condition and under local continuity and positivity condition Vilmos Totik (SZTE and USF) Polynomials 25 / *

133 Fine zero spacing Th. (Erdős-Turán, 1940) If w > 0 continuous on [ 1,1], then θ j 1 θ j π n θ j [ǫ,π ǫ] (2) No longer true if w can vanish! A form of universality in random matrix theory asymptotics for the kernel n p k (z +a/n)p k (z +b/n) a,b C k=0 Lubinsky, 2009: universality is true under the ρ Reg global condition and under local continuity and positivity condition Levin-Lubinsky: under these conditions (2) is true Vilmos Totik (SZTE and USF) Polynomials 25 / *

134 Universality fine zero spacing Vilmos Totik (SZTE and USF) Polynomials 26 / *

135 Universality fine zero spacing S the support of ρ, µ S the equilibrium measures of S Vilmos Totik (SZTE and USF) Polynomials 26 / *

136 Universality fine zero spacing S the support of ρ, µ S the equilibrium measures of S If I S is an interval, then dµ S (t) = ω S (t)dt Vilmos Totik (SZTE and USF) Polynomials 26 / *

137 Universality fine zero spacing S the support of ρ, µ S the equilibrium measures of S If I S is an interval, then dµ S (t) = ω S (t)dt Example: ω [ 1,1] (t) = 1 π 1 t 2 Vilmos Totik (SZTE and USF) Polynomials 26 / *

138 Universality fine zero spacing S the support of ρ, µ S the equilibrium measures of S If I S is an interval, then dµ S (t) = ω S (t)dt Example: ω [ 1,1] (t) = 1 π 1 t 2 Simon, T., 2010: ρ Reg and continuity and positivity of w(t) := dρ(t)/dt at an x Int(S) imply nω S (x)(x j+1 x j ) 1, x j x (3) Vilmos Totik (SZTE and USF) Polynomials 26 / *

139 Universality fine zero spacing S the support of ρ, µ S the equilibrium measures of S If I S is an interval, then dµ S (t) = ω S (t)dt Example: ω [ 1,1] (t) = 1 π 1 t 2 Simon, T., 2010: ρ Reg and continuity and positivity of w(t) := dρ(t)/dt at an x Int(S) imply nω S (x)(x j+1 x j ) 1, x j x (3) If log1/w L 1 (I), then (3) is true at almost all x I. Vilmos Totik (SZTE and USF) Polynomials 26 / *

140 Universality fine zero spacing S the support of ρ, µ S the equilibrium measures of S If I S is an interval, then dµ S (t) = ω S (t)dt Example: ω [ 1,1] (t) = 1 π 1 t 2 Simon, T., 2010: ρ Reg and continuity and positivity of w(t) := dρ(t)/dt at an x Int(S) imply nω S (x)(x j+1 x j ) 1, x j x (3) If log1/w L 1 (I), then (3) is true at almost all x I. Is (3)/universality true (say on [ 1, 1]) a.e. solely under the Erdős-Turán condition w > 0 a.e.? Vilmos Totik (SZTE and USF) Polynomials 26 / *

141 Critical points of polynomials Vilmos Totik (SZTE and USF) Polynomials 27 / *

142 Critical points of polynomials P n polynomial of degree n Vilmos Totik (SZTE and USF) Polynomials 27 / *

143 Critical points of polynomials P n polynomial of degree n z 1,...,z n zeros of P n, ξ 1,...,ξ n 1 zeros of P n Vilmos Totik (SZTE and USF) Polynomials 27 / *

144 Critical points of polynomials P n polynomial of degree n z 1,...,z n zeros of P n, ξ 1,...,ξ n 1 zeros of P n Gauss-Lucas, 1800 s: every ξ j is in the convex hull of {z 1,...,z n } Vilmos Totik (SZTE and USF) Polynomials 27 / *

145 Critical points of polynomials P n polynomial of degree n z 1,...,z n zeros of P n, ξ 1,...,ξ n 1 zeros of P n Gauss-Lucas, 1800 s: every ξ j is in the convex hull of {z 1,...,z n } Erdős-Niven, 1948: Vilmos Totik (SZTE and USF) Polynomials 27 / *

146 Critical points of polynomials P n polynomial of degree n z 1,...,z n zeros of P n, ξ 1,...,ξ n 1 zeros of P n Gauss-Lucas, 1800 s: every ξ j is in the convex hull of {z 1,...,z n } Erdős-Niven, 1948: 1 n 1 Iξ j 1 n 1 n 1 n Iz k 1 Vilmos Totik (SZTE and USF) Polynomials 27 / *

147 Critical points of polynomials P n polynomial of degree n z 1,...,z n zeros of P n, ξ 1,...,ξ n 1 zeros of P n Gauss-Lucas, 1800 s: every ξ j is in the convex hull of {z 1,...,z n } Erdős-Niven, 1948: 1 n 1 Iξ j 1 n 1 n 1 n Iz k 1 1 n 1 ξ j 1 n 1 n 1 n z k 1 Vilmos Totik (SZTE and USF) Polynomials 27 / *

148 Critical points of polynomials P n polynomial of degree n z 1,...,z n zeros of P n, ξ 1,...,ξ n 1 zeros of P n Gauss-Lucas, 1800 s: every ξ j is in the convex hull of {z 1,...,z n } Erdős-Niven, 1948: 1 n 1 Iξ j 1 n 1 n 1 n Iz k 1 de Bruijn-Springer: 1 n 1 ξ j 1 n 1 n 1 1 n 1 ξ j m 1 n 1 n 1 n z k 1 n z k m 1 Vilmos Totik (SZTE and USF) Polynomials 27 / *

149 Conjectures on majorization Vilmos Totik (SZTE and USF) Polynomials 28 / *

150 Conjectures on majorization de Bruijn-Springer conjecture (1948): if ϕ : C R + is convex, then 1 n 1 ϕ(ξ j ) 1 n 1 n 1 n ϕ(z k ) 1 Vilmos Totik (SZTE and USF) Polynomials 28 / *

151 Conjectures on majorization de Bruijn-Springer conjecture (1948): if ϕ : C R + is convex, then 1 n 1 ϕ(ξ j ) 1 n 1 n 1 n ϕ(z k ) 1 Very strong property, ( theory of majorization ; Weyl, Birkhoff, Hardy-Littlewood-Pólya) Vilmos Totik (SZTE and USF) Polynomials 28 / *

152 Conjectures on majorization de Bruijn-Springer conjecture (1948): if ϕ : C R + is convex, then 1 n 1 ϕ(ξ j ) 1 n 1 n 1 n ϕ(z k ) 1 Very strong property, ( theory of majorization ; Weyl, Birkhoff, Hardy-Littlewood-Pólya) Many other conjectures (by Schoenberg, Katroprinakis,...) on the relation of the z k s and ξ j s Vilmos Totik (SZTE and USF) Polynomials 28 / *

153 The Malamud-Pereira theorem An (n 1) n A = (a ij ) matrix is doubly stochastic if Vilmos Totik (SZTE and USF) Polynomials 29 / *

154 The Malamud-Pereira theorem An (n 1) n A = (a ij ) matrix is doubly stochastic if a ij 0, Vilmos Totik (SZTE and USF) Polynomials 29 / *

155 The Malamud-Pereira theorem An (n 1) n A = (a ij ) matrix is doubly stochastic if a ij 0, each row-sum equals 1 Vilmos Totik (SZTE and USF) Polynomials 29 / *

156 The Malamud-Pereira theorem An (n 1) n A = (a ij ) matrix is doubly stochastic if a ij 0, each row-sum equals 1 each column-sum equals (n 1)/n Vilmos Totik (SZTE and USF) Polynomials 29 / *

157 The Malamud-Pereira theorem An (n 1) n A = (a ij ) matrix is doubly stochastic if Let a ij 0, each row-sum equals 1 each column-sum equals (n 1)/n Z = z 1. z n Ξ = ξ 1. ξ n 1 Vilmos Totik (SZTE and USF) Polynomials 29 / *

158 The Malamud-Pereira theorem An (n 1) n A = (a ij ) matrix is doubly stochastic if Let a ij 0, each row-sum equals 1 each column-sum equals (n 1)/n Z = z 1. z n Ξ = ξ 1. ξ n 1 Th. (Malamud, Pereira, 2003) There is a doubly stochastic matrix A such that Ξ = AZ Vilmos Totik (SZTE and USF) Polynomials 29 / *

159 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: Vilmos Totik (SZTE and USF) Polynomials 30 / *

160 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: ξ j = k a jkz k Vilmos Totik (SZTE and USF) Polynomials 30 / *

161 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: ξ j = k a jkz k 1 n 1 j ϕ(ξ j ) 1 n 1 a jk ϕ(z k ) j k Vilmos Totik (SZTE and USF) Polynomials 30 / *

162 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: ξ j = k a jkz k 1 n 1 j = 1 n 1 ϕ(ξ j ) 1 n 1 ϕ(z k ) j k a jk ϕ(z k ) j k a jk = 1 ϕ(z k ) n k Vilmos Totik (SZTE and USF) Polynomials 30 / *

163 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: ξ j = k a jkz k 1 n 1 j = 1 n 1 ϕ(ξ j ) 1 n 1 ϕ(z k ) j k a jk ϕ(z k ) j k a jk = 1 ϕ(z k ) n k Examples: 1) 1 n 1 Rξ j m 1 n 1 n 1 n Rz k m 1 Vilmos Totik (SZTE and USF) Polynomials 30 / *

164 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: ξ j = k a jkz k 1 n 1 j = 1 n 1 ϕ(ξ j ) 1 n 1 ϕ(z k ) j k a jk ϕ(z k ) j k a jk = 1 ϕ(z k ) n k Examples: 1) 1 n 1 Rξ j m 1 n 1 n 1 n Rz k m 1 2) If all zeros lie in the upper-half plane, then Vilmos Totik (SZTE and USF) Polynomials 30 / *

165 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: ξ j = k a jkz k 1 n 1 j = 1 n 1 ϕ(ξ j ) 1 n 1 ϕ(z k ) j k a jk ϕ(z k ) j k a jk = 1 ϕ(z k ) n k Examples: 1) 1 n 1 Rξ j m 1 n 1 n 1 n Rz k m 1 2) If all zeros lie in the upper-half plane, then ( n ) 1/n Iz k 1 ( n 1 1 Iξ j ) 1/(n 1) Vilmos Totik (SZTE and USF) Polynomials 30 / *

166 Gauss-Lucas, de Brjuin-Springer etc. are all immediate consequences: ξ j = k a jkz k 1 n 1 j = 1 n 1 ϕ(ξ j ) 1 n 1 ϕ(z k ) j k a jk ϕ(z k ) j k a jk = 1 ϕ(z k ) n k Examples: 1) 1 n 1 Rξ j m 1 n 1 n 1 n Rz k m 1 2) If all zeros lie in the upper-half plane, then ( n ) 1/n Iz k 1 ( n 1 1 Iξ j ) 1/(n 1) Vilmos Totik (SZTE and USF) Polynomials 30 / *