Root statistics of random polynomials with bounded Mahler measure

Transcription

1 Root statistics of random polynomials with bounded Mahler measure Maxim L. Yattselev Indiana University-Purdue University Indianapolis AMS Sectional Meeting Bloomington, IN April 1st, 2017

2 Mahler Measure The Mahler measure of a polynomial P(z) = a N n=1 (z αn) is defined as N M(P) := a max { 1, α } n = exp n=1 { 1 2π 2π 0 log P ( e iθ) } dθ.

3 Mahler Measure The Mahler measure of a polynomial P(z) = a N n=1 (z αn) is defined as N M(P) := a max { 1, α } n = exp Theorem (Kronecker, 1857) n=1 { 1 2π 2π 0 log P ( e iθ) } dθ. M(P) = 1 for a polynomial P with integer coefficients iff P is a product of monomials and cyclotomic polynomials (divisors of z n 1). Necessarily, such a polynomial has all its roots in T {0}.

4 Mahler Measure The Mahler measure of a polynomial P(z) = a N n=1 (z αn) is defined as N M(P) := a max { 1, α } n = exp Theorem (Kronecker, 1857) n=1 { 1 2π 2π 0 log P ( e iθ) } dθ. M(P) = 1 for a polynomial P with integer coefficients iff P is a product of monomials and cyclotomic polynomials (divisors of z n 1). Necessarily, such a polynomial has all its roots in T {0}. Conjecture (Lehmer, 1933) Is 1 an isolated point of the range of M( ) on integer polynomials? Lehmer himself constructed the smallest known example: M(z 10 + z 9 z 7 z 6 z 5 z 4 z 3 + z + 1) 1.18.

5 Number of Integer Polynomials Theorem (Chern-Vaaler, 2001) The number of integer polynomials of height at most T behaves as vol ( B N ) T N+1 + O(T N ), T, where B N is the Mahler measure unit star body. Moreover, where and C N is an explicit constant. vol ( B N ) = 2 N + 1 F N (N + 1), (N 1)/2 F N (s) = C N m=0 s s (N 2m) Notice that Lehmer s conjecture asks what happens when T 1. Observe also that both (z 1) N and z N 1 belong to B N but have drastically different coefficient vectors.

6 Volumes of Star Bodies More generally, the λ-homogeneous Mahler measure is given by M λ (P) = a λ N n=1 max { 1, α n }. The corresponding unit star body is defined as { ( N ) } BN λ := (a 1,..., a N+1 ) R N+1 : M λ a n+1z n 1. n=0 Theorem (Chern-Vaaler, 2001) vol ( BN λ ) 2 = N + 1 F N ( N + 1 λ ).

7 Volumes of Star Bodies vol ( BN λ ) = = ( ) } N 1 vol {b : M λ cz N + b n+1z n 1 dc n=0 ( ) } N 1 vol {cb : M λ cz N + cb n+1z n 1 dc. Using the λ-homogeneity of M λ one then gets vol ( BN λ ) { = 2 c N vol b : M(b) c λ} dc = 2 λ 0 0 n=0 ξ (N+1)/λ vol { b : M(b) ξ } dξ ξ, where M(b) is the Mahler measure of z N + N 1 n=0 bn+1z n. Integration by parts then gives vol ( BN λ ) 2 = ξ (N+1)/λ dvol { b : M(b) ξ } N = M(b) (N+1)/λ dµ N R (b). N + 1 R N

8 Volumes of Star Bodies Making a change of variables from coefficients of polynomials to their roots gives F N (s) := M(b) s dµ N R (b) = Z L,M (s), R N L!M! L+2M=N where L and M stand for the number of real and complex roots, and Z L,M (s) = R L C M l=1 L Φ(α l ) s M m=1 Φ(β m) 2s (α, β) dµ L R(α)dµ M C (β) with (α, β) being the Vandermonde of α 1,..., α L, β 1, β 1,..., β M, β M. The summands Z L,M (s) are not simple and Chern-Vaaler went through a dozen pages of rational function identities to show that (N 1)/2 F N (s) = C N m=0 s s (N 2m).

9 Complex Star Bodies In fact, one could consider polynomials with complex coefficients. Set { ( N ) } BN λ (C) := (a 1,..., a N+1 ) C N+1 : M λ a n+1z n 1. Then vol ( B λ N (C) ) = = = = C C n=0 ( ) } N 1 vol {b : M λ cz N + b n+1z n 1 dc n=0 { c 2N vol b : M(b) c λ} dc π ξ 2(N+1)/λ dvol { b : M(b) 1 } N π M(b) 2(N+1)/λ µ N C (b). N + 1 C N

10 Complex Star Bodies As before, making a change of variables from the coefficients to the roots gives where Z N (s) = G N (s) := M(b) 2s dµ N C (b) = Z N(s), C N N! C N n=1 = (2π) N σ = (2π) N σ N Φ(λ n) 2s (λ) 2 dµ N C (λ) ( N n=1 ( N n=1 0 ) Φ(ρ n) 2s ρn 2σ(n) 1 dρ n ) s 2σ(n)(s σ(n)) = π N N n=1 s s n. Theorem (Chern-Vaaler, 2001) vol ( B λ N (C) ) = π N + 1 G N ( N + 1 λ ).

11 Determinantal Interpretation Theorem (Sinclair, 2008) Let Π 0,..., Π N 1 be polynomials such that Π n Π m = δ n,m, where inner product is defined by f g C = f (z)g(z)φ(z) 2s dµ C (z), with Φ(z) := max{1, z }. Then where Π k (z) = γ k z k +. C G N (s) = N 1 n=0 γ 2 n,

12 Pfaffian Interpretation Theorem (Sinclair, 2008) Let π 0,..., π N 1 be polynomials such that π 2n π 2m = π 2n+1 π 2m+1 = 0 and π 2n π 2m+1 = δ n,m, where skew-symmetric inner product = R + C is defined by f g R = f (x)g(y)sgn(y x)φ(x) s Φ(y) s dµ R (x)dµ R (y) R R f g C = 2i f (z)g(z)sgn(im(z))φ(z) 2s dµ C (z), with Φ(z) = max{1, z }. Then where π k (z) = γ k z k +. C F N (s) = (N 1)/2 n=0 ( γ2nγ 2n+1 ) 1,

13 Random Polynomials Recall that G N (s) = M(b) 2s dµ N C (b) and F N (s) = M(b) s dµ N R (b). C N R N Under a random polynomial we mean a polynomial chosen with respect to M(b) 2s /G N (s), b C N, or M(b) s /F N (s), b R N. This is equivalent to choosing polynomials uniformly from B (N+1)s 1 N. We would like to study fine statistics of zeros of such random polynomials.

14 Numerical Simulation A simultaneous plot of the roots of 100 random polynomials of degree 28. A ball-walk of 10, 000 steps of length.01 starting from x 28 was performed for each polynomial. The arrows indicate directions of outlying roots.

15 Correlation Functions: Complex Case Let P be a random polynomial. For C C define N C := C {zeros of P}. In the case of complex coefficients, a function R n : C n [0, ) is called n-th correlation function if E[N C1 N Cn ] = R n(z)dµ n C(z) C 1 C n for pairwise disjoint sets C 1,..., C n. Since the joint density of the zeros is given by 1 N λ n λ m 2 Φ(λ n) 2s dµ N C (λ), Z N (s) m<n n=1 Φ(z) = max{1, z }, it is well known in random matrix theory that where R n(λ) = det [ K N (λ i, λ j ) ] n i,j=1, N 1 K N (z, w) := Φ(z) s Φ(w) s Π n(z)π n(w) and Π n are orthonormal polynomials w.r.t. Φ 2s (z)dµ C (z). n=0

16 Correlation Functions: Real Case In the case of real coefficients, if there is a function R l,m : R l C m + [0, ) such that E [ ] N A1 N Al N B1 N Bm := R l,m (x, z)dµ A l R(x)dµ m C (z) 1 B 1 B m for pairwise disjoint sets A 1,..., A l R and B 1,..., B m C +, then it is called the (l, m)-th correlation function. A l When such functions exist, it holds in particular that deg(p) = R 1,0(x, )dµ R (x) + R 0,1(, z)dµ C (z) R and the first integral represents the expected number of real zeros, where we set R l,m (, z) := R l,m (, z). C

17 Correlation Functions: Real Case Theorem (Borodin-Sinclair, 2009) There exists a 2 2 matrix kernel K N : C C C 2 2 such that [ K N (x i, x j ) ] l [ i,j=1 K N (x i, z ] l,m n) i,n=1 R l,m (x, z) = Pf In particular, it holds that [ K T N (z k, x j ) ] m,l k,j=1 [ K N (z k, z n) ] m k,n=1 R 1,0(x, ) = PfK N (x, x) and R 0,1(, z) = PfK N (z, z).. Recall that we set = R + C, where f g R = f (x)g(y)sgn(y x)φ(x) s Φ(y) s dµ R (x)dµ R (y) R R f g C = 2i f (z)g(z)sgn(im(z))φ(z) 2s dµ C (z). C

18 Correlation Functions: Real Case Theorem (Borodin-Sinclair, 2009) Let N = 2J and π 0,..., π N 1 be skew-orthogonal polynomials w.r.t.. Set Then κ N (u, v) := 2Φ(u) s Φ(v) s K N (u, v) = J ( π2n(u)π 2n+1(v) π ) 2n(v)π 2n+1(u). n=0 κ N(u, v) κ N ɛ(u, v), ɛκ N (u, v) ɛκ N ɛ(u, v) + 1 sgn(u v) 2 where sgn( ) = 0 for non-real arguments and ɛ is the operator ɛf (u) := 1 f (t)sgn(t u)dµ 2 R R(t), u R, i sgn(im(u))f (u), u C \ R, which acts on u when written on the left and on v when written on the right.

19 Skew-Orthogonal Polynomials The following results are from Sinclair-Ya (complex case) and 2015 (real case). Theorem It holds that and π 2n(z) = 2 π π 2n+1(z) = 1 2π It is also true that n k=0 n k=0 Π n(z) = Γ(k + 3/2)Γ(n k + 1/2) z 2k Γ(k + 1)Γ(n k + 1) s (2k + 2) 2s Γ(k + 3/2)Γ(n k 1/2) z 2k+1. Γ(k + 1)Γ(n k + 1) ( n n + 1 ) z n. π s

20 Expected Number of Real Zeros Write π k (z) := π k (z)φ(z) s. Given A R and N even, it holds that E[N A ] = PfK N (x, x)dµ R (x) A [ ] 0 κ N ɛ(x, x) = Pf dµ R (x) ɛκ N (x, x) 0 A N/2 = 2 ( π2n(x)ɛ π 2n+1(x) π ) 2n+1(x)ɛ π 2n(x) dµ R (x). n=0 A

21 Expected Number of Real Zeros Theorem Let N in and N out be the number of real roots on [ 1, 1] and R \ ( 1, 1). Then E[N in ] = 1 π log N + O N(1) E[N out] = 1 N(2s N) log ( 1 Ns 1) + Ns π s 1 O N (1), where the implicit constants are uniform with respect to s. Observe that E[N out] = Ns 1 O N (1), lim sup N Ns 1 < 1, α π log N + O N(1), s = N + N 1 α, α [0, 1], 1 π log N + O N(1), lim sup N (s N) <.

22 Expected Number of Zeros Around Points of the Unit Circle Let ζ T and δ be small. In the complex case we have that E [ ] N ζ+δd = K N (z, z)dµ C (z) ζ+δd = δ 2 ( ) K N ζ + δz, ζ + δz dµc (z). Similarly, in the real case we have for ζ T \ {±1} that E [ ] N ζ+δd = PfK N (z, z)dµ C (z). D ζ+δd As we have N total zeros, the scale should be δ = 1/N.

23 Expected Number of Zeros Around Points of the Unit Circle Theorem Let ζ T. Assume that λ := lim N Ns 1 [0, 1] exists. Then 1 ( lim N N K 2 N ζ + z N, ζ + w ) = K ζ (z, w), N } where ω(τ) := min {1, e Re(τ)/λ and K ζ (z, w) = ω ( zζ ) ω ( wζ ) 1 π 1 It holds that Re(zζ) > 0 iff z points outside D at ζ. 0 x(1 λx)e (zζ+wζ)x dx.

24 Expected Number of Zeros Around Points of the Unit Circle Theorem Let ζ T \ {±1}. Assume that λ := lim N Ns 1 [0, 1] exists. Then [ ] 1 ( lim N N K 2 N ζ + z N, ζ + w ) 0 K ζ (z, w) =, N K ζ (w, z) 0 That is, Pfaffian point process becomes essentially determinantal around ζ.

25 Expected Number of Zeros Around Points of the Unit Circle Theorem Let ζ T \ {±1}. Assume that λ := lim N Ns 1 [0, 1] exists. Then [ ] 1 ( lim N N K 2 N ζ + z N, ζ + w ) 0 K ζ (z, w) =, N K ζ (w, z) 0 That is, Pfaffian point process becomes essentially determinantal around ζ. Theorem Let ξ {±1}. Assuming that λ := lim N Ns 1 [0, 1] exists, it holds that lim N 1 ( N κ 2 N x + u N, ξ + v ) = κ ξ (u, v) N where the convergence is locally uniform in C C, κ ξ (u, v) = ω (uξ) ω (vξ) ξ 1 ( ) τ(1 λτ) M (uξτ)m(vξτ) M(uξτ)M (vξτ) dτ, 4 0 and M(z) = 1F 1(3/2, 1; z), i.e., zm (z) + (1 z)m (z) 3 M(z) = 0. 2

26 Expected Number of Zeros Around ±1 The scaled intensity of complex roots near 1, for λ = 1 (left) and λ = 0 (right). Note how the roots tend to accumulate near the unit disk (the y -axis here) and repel from the real axis.

27 Expected Number of Zeros on Compact Subsets of D Theorem Assuming that λ := lim N Ns 1 [0, 1] exists, it holds that and lim K N(z, w) = 1 1 N π (1 zw) 2 lim κ N(u, v) = 1 N 4π T ( v τ u τ ) dτ ( 1 u2 τ ) 3/2( 1 v 2 τ ) 3/2 locally uniform in D D, where τ is the branch defined by 2 π τ m 2m 1.

28 Expected Number of Zeros on Bounded Subsets of C \ D Theorem Assuming that λ := lim N Ns 1 [0, 1] amd c := lim N (s N) [0, ] exist, it holds that zw s K N (z, w) lim N (zw) N s N = λ ] 1 [1 + c 1 π zw 1 zw 1 and lim N uv s κ N (u, v) (uv) N s N = λ π ] 1 [1 + c 1 uv 1 uv 1 v u u2 1 v 2 1.

29 Expected Number of Zeros on Bounded Subsets of C \ D The limiting intensity of complex roots outside the disk, with a close up view near z = 1, for the Mahler measure (c = 1) case.