RESEARCH STATEMENT TURGAY BAYRAKTAR

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1 RESEARCH STATEMENT TURGAY BAYRAKTAR My research centers around complex analysis, complex geometry and probability theory. In one direction, I combine pluripotential theory with modern complex geometry techniques to study ergodic properties of holomorphic endomorphisms of complex manifolds and classification of varieties admitting dynamically interesting maps. From the probabilistic point of view, I am interested in stochastic properties of dynamical systems generated by random iteration of holomorphic endomorphisms. In the past few years, my focus has shifted toward random algebraic geometry, which explores local and global statistical properties of algebraic varieties when the coefficients in defining polynomials are considered as random variables. Below, I outline motivating sources behind my research projects, explain their achievements and describe future goals.. Universality of random zeros In the most general sense a random object of any kind means putting a probability measure on the set of objects and consider them as elements of a probability space. Instead of studying the individuals, one focuses on probabilities, averages and correlations. Here, we focus on global holomorphic sections H 0 (X, L n ) of tensor powers of a positive line bundle L X defined over a projective manifold X. For a fixed basis of H 0 (X, L n ), we consider linear combinations in which coefficients are non-degenerate i.i.d. real or complex random variables. The present geometric setting is a natural generalization of random polynomials with i.i.d. coefficients relative to a basis (eg. monomials, orthogonal polynomials). The basic questions are concerned with asymptotic distribution and correlations between zeros of generic, in the sense of probability, sequences of sections as the degree n. Motivating sources are coming from various areas: In complex analysis value distribution theory of polynomials or analytic functions is a classical topic but computable examples may not exhibit the typical behavior. One can study expected distribution of zeros, almost sure convergence of them etc. to understand the typical distribution. Moreover, random polynomials serve as a model for eigenfunctions of quantum chaotic Hamiltonians [BBL96]. Another motivation comes from algebraic geometry: If X = P k is the complex projective space and L = O() is the hyperplane bundle then holomorphic sections in H 0 (X, L n ) are essentially homogenous polynomials of degree n. In particular, simultaneous zero set Z s n,...,s k of k generic sections n defines an algebraic submanifold of codimension k and one would like to understand typical asymptotic behavior of such objects. For a smooth positively curved metric h on L X we denote its curvature form by ω h which is a positive smooth closed (, ) form on X and the induced volume form by dv := m! ωm h. Then there exists an associated scalar L2 -norm on the finite dimensional Hilbert space H 0 (X, L n ) and we denote the unit sphere with respect to this norm by SH 0 (X, L n ). Shiffman and Zelditch [SZ99, SZ08] studied asymptotic zero distribution of spherical ensemble SH 0 (X, L n ) endowed with U(d n )-invariant probability measure where d n = dim(h 0 (X, L n )). They proved that for a random sequence s n SH 0 (X, L n ) the Date: September 2, 205.

2 2 TURGAY BAYRAKTAR normalized zero currents n [Z n] converge almost surely to curvature form ω h as n. In particular, if the sections s n are chosen independently and at random then the zeros are uniformly distributed. In fact, for this example an implicit choice of an inner product on H 0 (X, L n ) forces zeros to distribute uniformly. More precisely, this ensemble can be equivalently obtained by fixing an orthonormal basis relative the L 2 -norm on H 0 (X, L n ) and letting the random coefficients to be i.i.d. standard complex Gaussians. Then several questions arise. Are zeros of random holomorphic sections always asymptoticly uniformly distributed? What are the geometric reasons behind all this that force random zeros asymptoticly equidistributed? Do non-gaussian ensembles of random holomorphic sections have the same asymptotic zero distribution as the Gaussian ensemble? The answer of the first two questions is related to the choice of the scalar inner product on H 0 (X, L n ) and the corresponding Bergman kernel asymptotics whereas for the last one, under mild assumptions on the distribution law of the coefficients we expect to have the asymptotic zero distribution to be independent of its choice. The universality phenomenon in the context of global holomorphic sections roughly says that as n asymptotic distribution of (appropriately normalized) zeros of random holomorphic sections should become independent of the choice of distribution of random coefficients. In [Bay3a, Bay4] I proved universality phenomenon (in any codimension) at macroscopic (global) scales for heavy tailed distributions. More recently [Bay5b], in complex dimension one it is proved that a mild log-integrability condition on the distribution of random coefficients is necessary and sufficient for universality of counting measures of zeros. In the sequel, I will describe details of these results and provide some examples to illustrate them. For a regular compact set K X and a continuous weight function Q : K R there is a naturally defined pluri-potential theoretic extremal function V K,Q. Let us denote equilibrium current by T K,Q := ω h + i π V K,Q. We remark that if K = X and Q 0 then T K,Q coincides with the curvature form ω h and we recover Shiffman-Zelditch setting. Another interesting example is that if K C is a regular compact set and Q 0, then T K,Q is the equilibrium measure of K which is the unique minimizer of the logarithmic energy functional ν log z w dν(z)dν(w) over all probability measures supported on K. For a fixed related orthonormal basis, we assume that random coefficients a j are i.i.d random variables whose distribution law P has a bounded density and logarithmically decaying tails i.e. P{a j C : log a j > R} = O(R ρ ) as R for some ρ > m +. Then it follows from Bertini s theorem that with probability one zero loci of random holomorphic sections intersect transversally. The following result describes asymptotic expected and almost sure distribution of normalized simultaneous zero sets. Theorem.. [Bay3a] Let X be a projective manifold and L X be a positive line bundle then for each k dim C X n k E[Z s n,...,sk n ] T k K,Q in the sense of currents as n. Moreover, if the ambient space X is complex homogeneous then almost surely n k [Z s n,...,s k n ] T k K,Q

3 in the sense of currents as n. RESEARCH STATEMENT 3 Recall that complex projective space P m is among the examples of complex homogenous manifolds. In particular, many classical random complex polynomial ensembles arise as a special case of Theorem.. For instance, if K = S is the unit circle in the complex plane then monomials form an ONB for polynomials ralative to the inner product f, g = f(ζ)g(ζ)dζ 2π S and our setting reduce to Kac random polynomials f n (z) = n j= a jz j the current of integration [Z fn ] = {z:f δ n(z)} z is the empirical measure of zeros and the equilibrium measure of S is the normalized angular measure 2π dθ. In particular, Theorem. implies that normalized zeros of Kac random polynomials of large degree tend to accumulate on the unit circle S and we recover a classical result due to Kac and Hammersley [Kac43, Ham56]. This ensemble of random polynomials has been studied extensively among others by [LO43, HN08, SV95] (see also [SZ03, Blo07, BS07, BL3] for more general random polynomial ensembles). In the univariate case, universality of zeros of random Kac polynomials was obtained by Ibragimov and Zaporozhets [IZ3] under a mild log-integrability condition. In a recent work [Bay5b], I generalized their result to the setting of random holomorphic sections. Theorem.2. [Bay5b] The logarithmic moment (.) E[log( + a j )] < if and only if { } (.2) P rob s n : lim n n V olω h 2m 2 (Z s n U) = V U = for every open set U X such that U has zero Lebesgue measure. Here, we denote the volume of the hypersurface Z sn in an open set U with respect to the volume form dv by V ol ω h 2m 2 (Z s n U) = ω m (m )! h and V U = T K,Q ω m Z sn U (m )! h. U In complex dimension one, V ol 2m 2 (Z sn U) becomes counting measure N n (s n, U) of zeros of s n in U. In particular, for an appropriate choice of the inner product on H 0 (X, L n ), Theorem.2 provides a necessary and sufficient condition for universality of elliptic polynomials (also known as SU(2) polynomials [BBL96]). Another interesting ensemble arising in [Bay5b] is Weyl polynomials. Recall that a random Weyl polynomial is of the from W n (z) = n j=0 a j n j j! z j. In [Bay5b], it is proved that condition (.) holds if and only if almost surely n N n(w n, U) λ(u D) n where λ is the Lebesgue measure on C and D denote the unit disc in the complex plane. Thus, we obtain an analogue of the circular law for random matrices (cf. [BC2]) in the present setting. For instance, Figures and 2 below illustrate the similar appearance of the zero distribution of a random Weyl polynomial of degree 2000 with i.i.d. Gaussian and Bernoulli (taking values ± with probability 2 ) random coefficients respectively.

4 4 TURGAY BAYRAKTAR Figure. Gaussian Figure 2. Bernoulli.. Random sparse polynomials. Recall that Newton polytope of a multivariable polynomial f(z) = a J z J is the convex hull of its support S f = {J Z m : a J 0}. It is well known that Newton polytope of a polynomial has crucial influence on its value distribution in particular on its zero set. A well-known theorem of Bernstein-Kouchnirenko asserts that the number of zeros of (a generic family of) m polynomials with Newton polytope P is m!v ol(p ). There is a vast literature on sparse polynomials (see eg. [HS95, Roj96, MR04, DGM3]) which are using methods of algebraic geometry to study zero distribution of systems of such polynomials. More recently, limiting zero distribution of Gaussian random sparse polynomials (induced from SU(m + ) polynomials) is studied by Shiffman and Zelditch [SZ04] from the complex analytic point-view. In [Bay4], I develop a pluri-potential theory to study asymptotic zero distribution of more general random sparse polynomial ensembles whose support are contained in dilates of a fixed integral polytope P as their degree grow. The universality phenomenon in the present setting is established by obtaining a quantitative localized version of Bernstein-Kouchnirenko Theorem [Bay4]..2. Work in progress. Another interesting topic related to random polynomials is the study of local statistics of their zero sets. In [ZZ0], Zelditch and Zeitouni proved that the normalized empirical measures of zeros of Gaussian random polynomials in one complex variable satisfies a large deviation principle (LDP). In a forthcoming project [Bayb], I consider hole probabilities for zeros of random polynomials and obtain moderate and large deviations estimates for non-gaussian ensembles. In particular, these results imply that there is no hope for LDP for heavy tailed distributions unless they have exponentially decaying tails. On the other hand, it is a questions that if LDP holds for distributions with exponentially decaying tails (eg. sub-gaussians). LDP for random polynomials with exponentially distributed coefficients was established in a more recent paper [GZ3]. I am interested in generalizing this LDP result to the setting of distributions with exponentially decaying tails. More generally, I intend to formulate a LDP in the case of random polynomials of several complex variables. Another universality form for zeros of random holomorphic sections is central limit theorem (CLT) for linear statistics of zeros. That is for a given smooth (m, m ) test form ϕ we consider the random variable [Z sn ], ϕ. In [Baya], we establish asymptotic normality of the random variables [Z sn ], ϕ E[ [Z sn ], ϕ ] V ar [Zsn ], ϕ for sufficiently smooth weight function Q. In particular, these results generalize [ST04, SZ0].

5 RESEARCH STATEMENT 5 2. Holomorphic Dynamics 2.. Ergodic and stochastic properties of holomorphic maps. A dynamical system, in the most general sense, is an object that evolves in time according to a fixed mathematical rule. In particular, a rational map f : P P of the Riemann sphere defines a discrete dynamical system by iteration. The dynamical study of rational maps of the Riemann sphere was initiated by Fatou and Julia in the early 900 s and has developed into a very rich and beautiful subject. There is a dichotomy between the Julia set where the dynamics is chaotic and the Fatou set on which the dynamics is predictable. A basic tool in the iteration theory of rational maps of P is Montel s theorem on normal families. In higher dimensions, however, the analogues of Montel s theorem do not have the appropriate adaptability. On the other hand, pluripotential theory reveals itself as a main tool in the multi-dimensional case. Following the successful study of rational maps of the Riemann sphere a great deal of research has gone into theory of iteration of rational maps in higher dimensions. Various categories of maps have been studied within the past two decades, particularly holomorphic maps of complex projective space P k (see [Sib99] and references therein). For a holomorphic endomorphism f : P k P k of algebraic degree d 2, its Green current T f is defined to be the weak limit of the sequence of smooth forms {d n (f n ) ω F S } where ω F S denotes Fubini-Study form on P k. Dynamical Green currents play a crucial role in the study of the iteration of holomorphic maps of P k. For instance, the support of the Green current is the Julia set. Moreover, T f has Hölder continuous quasi-potentials. Therefore, by Bedford-Taylor theory one can define the exterior powers of T f which are also dynamically interesting currents. In particular, the top degree self-intersection µ f = T f T f yields the unique f-invariant measure of maximal entropy [FS95, Sib99, BD0]. Statistical properties of µ f have also been extensively studied. By means of different methods it was proved that µ f satisfies Central Limit Theorem (CLT for short) for Hölder continuous observables ψ : P k R [CLB05, Dup0, DNS0] and for dsh observables (i.e. differences of quasi-plurisubharmonic functions) [DNS0]. Motivated by the deterministic case, in [Bay5a] I considered ergodic properties of composition of holomorphic endomorphisms f n f f 0 where f j s are independently chosen at random according to some probability distribution. More precisely, the set of rational endomorphisms f : P k P k with fixed algebraic degree d 2 can be identified with complex projective space P N for some N N and the set of holomorphic endomorphisms coincides with a Zariski open set H d P N. We fix a Borel probability measure P on P N and assume that a random rational map is holomorphic with probability one. By a random holomorphic endomorphism we mean a P N -valued random variable whose distribution law is P. For a sequence λ := (f 0, f,... ) of random holomorphic maps we denote F λ,n := f n... f f 0 and θ(λ) = (f, f 2,... ). Note that if m is a Dirac mass supported at f H d then the deterministic case emerges. On the other hand, an atomic measure of the form m = r r j= δ f j with f j H d gives rise to iterations of holomorphic maps in a finitely generated semi group. The next result indicates that almost surely the sequence of pullbacks of a smooth measure by a random sequence of holomorphic maps is equidistributed with a measure which depends only on the sequence: Theorem 2.. [Bay5a] If (2.) log dist(, P N \H d ) L m(p N )

6 6 TURGAY BAYRAKTAR then there exists a θ-invariant set A Hd N of probability one such that for almost every λ = (f 0, f,... ) A, the sequence of smooth forms {d pn Fλ,n ωk F S } converge weakly to a measure µ(λ) satisfying f0 (µ(θ(λ))) = d k µ(λ). Moreover, the measure µ(λ) has Hölder continuous potentials. The assumption (2.) holds in a quite general setting for instance, it holds any volume form on P N. Then we consider the expected value of the measure-valued random variable µ(λ) which is a probability measure whose action on a continuous function φ : Hd N Pk R is given by µ, φ := µ(λ), φ(λ, ) dp(λ) H N d where P is the product measure induced by P. The expected measure µ plays the role of the measure of maximal entropy defined in the deterministic case. Namely, µ is invariant under the natural skew product τ : H N d Pk H N d Pk τ(λ, x) = (θ(λ), f 0 (x)) i.e. τ µ = µ and the dynamical system (H N d Pk, µ, τ) is mixing and µ maximizes the mixed entropy introduced by Abramov and Rohlin [Bay5a, DT5]. Next, we turn out attention to statistical properties of the measure µ. The dynamical system (H N d Pk, µ, τ) has exponential decay of correlations for Hölder continuous and d.s.h. observables whenever the support of m is contained in H d [Bay5a]. The later result also allows us to obtain CLT for Hölder continuous and dsh observables. This generalizes the corresponding result of Dinh, Nguyen and Sibony [DNS0] proved for the deterministic case and establishes a universality result in the setting of random dynamical systems. Theorem 2.2. [Bay5a] If supp(m) H d then for every Hölder continuous function ψ : P k R which is not a coboundary and µ, ψ π = 0 there exists σ > 0 such that lim µ{ (λ, x) : n for every interval I R. n n j=0 ψ(f λ,j (x)) I } = 2πσ I exp( t2 2σ 2 )dt 2.2. Dynamics of meromorphic maps. Let f : P P be a rational map of degree d 2 and ν be any probability measure on P. A theorem of Brolin [Bro65], Lyubich [Lju83] and Freire-Lopez-Mañé [FLM83] asserts that the pre-images d n (f n ) ν converge weakly to the measure of maximal entropy µ f if and only if ν(e f ) = 0 where E f is an (possibly empty) exceptional set. In the graduate school, I considered the above equidistribution problem in a more general setting [Bay3c, Bay3b]. Let X be a compact Kähler manifold of complex dimension k and f : X X be a meromorphic endomorphism whose range contains an open set. In higher dimensions, meromorphic maps are not necessarily continuous, due to the possible existence of points of indeterminacy. The term map is loosely used here. Therefore, many natural concepts such as pull-back of a smooth form by f, topological (or metric) entropy etc. that are defined for smooth maps, require additional interpretation in the meromorphic setting. Although a meromorphic map is not smooth it induces a linear action on the cohomology f : H, (X, R) H, (X, R).

7 RESEARCH STATEMENT 7 Specifically, we focus on meromorphic maps with exponential growth on cohomology and study the existence of invariant currents representing the cohomology classes which are expanded under pull-back. Unlike the rational maps of P, there is no general existence theorem for the Green currents nor a complete characterization of the exceptional set of positive closed currents on compact Kähler manifolds. In this new setting, we replace smooth measures with smooth forms of bidegree (, ) and Dirac masses at a point with current of integrations along hypersurfaces. A natural question in dynamics is to provide necessary and sufficient conditions on a smooth form or more generally a positive closed current S such that the sequence of pull-backs {λ n f (f n ) S} n converges weakly to the Green current T f where λ f denotes the first dynamical degree of f. In [Bay2, Bay3c, Bay3b], I provided existence theorems for an f -invariant extremal current T f which has good convergence properties. For instance, in the algebraic case if the cohomology class of T f is the first Chern class of a finitely generated big line bundle then the Green current describes the limiting distribution of pre-images of zero divisors of global holomorphic sections [Bay3b]. In particular, if X = P k and L is the hyperplane bundle O() then we recover the corresponding result of Russakovskii and Shiffman [RS97] Automorphism groups of compact Kähler manifolds. A holomorphic automorphism of a compact Kähler manifold f : X X induces a linear action on the cohomology H (X, C) by pulling-back smooth forms. This provides a morphism Aut(X) GL(H (X, Z)) and we denote the image of this morphism by Aut(X). The notion of entropy is used to measure complexity of a dynamical system. Intuitively, the bigger entropy implies a richer dynamics. It follows from a theorem of Gromov and Yomdin that f Aut(X) has positive topological entropy if and only if one of the eigenvalues of f has modulus larger than one. In his works [Can99, Can0], Cantat proved that if X is a compact complex surface admitting a holomorphic automorphism with positive entropy then X is obtained from the complex projective plane P 2, a torus, a K3 surface or an Enriques surface by a finite sequence of point blow-ups. In particular, if X is a rational surface admitting an automorphism with positive entropy then it is obtained from P 2 by blowing up a finite sequence of at least ten points [Nag6]. Examples of rational surface automorphisms with positive entropy were given by McMullen [McM07], Bedford and Kim [BK0]; these examples are obtained from birational transformations f of the complex projective plane P 2 by a finite sequence of blow-ups that resolve all indeterminacies of f and f simultaneously. However, in higher dimensions it is not known that if one can obtain automorphisms with interesting dynamics on rational manifolds by blowing up along smooth centers. In a joint work with S. Cantat, we addressed this question in certain cases: Theorem 2.3. [BC3] Let X 0 be a smooth, connected, complex projective variety with Picard number (respectively compact Kähler manifold with h, (X 0 ) = ), m be a positive integer, and π i : X i+ X i, i = 0,..., m, be a sequence of blow-ups of smooth irreducible subvarieties of dimension at most r. If dim(x 0 ) > 2r + 2 then Aut(X m ) is a finite group. An immediate consequence is that every automorphism of such X m has zero topological entropy. In particular, if X is obtained from X 0 = P k with k 3 by blowing up a finite sequence of points then every holomorphic automorphism of X has zero topological entropy (see also [Bay3d]).

8 8 TURGAY BAYRAKTAR References [Baya] T. Bayraktar. Asymptotic normality of fluctuations of zeros of random holomorhic sections. in preparation. [Bayb] T. Bayraktar. Hole probabilities for zeros of random polynomials: Large and moderate deviations. in preparation. [Bay2] T. Bayraktar. Green currents for meromorphic maps of compact Kähler manifolds. PhD thesis, Indiana University, 202. [Bay3a] T. Bayraktar. Equidistribution of zeros of random holomorphic sections. to appear in Indiana Univ. Math. J., arxiv , 203. [Bay3b] T. Bayraktar. Equidistribution toward the green current in big cohomology classes. Internat. J. Math., 24(0):25pp, 203. [Bay3c] T. Bayraktar. Green currents for meromorphic maps of compact kähler manifolds. J. Geom. Anal., 23(2): , 203. [Bay3d] T. Bayraktar. On the automorphism group of rational manifolds. unpublished manuscript, arxiv:20.465, 203. [Bay4] T. Bayraktar. Zero distribution of random polynomials with fixed newton polytope. submitted to Trans. Amer. Math. Soc., arxiv , 204. [Bay5a] T. Bayraktar. Ergodic properties of random holomorphic endomorphisms of P k. Int. Math. Res. Not. IMRN, (4): , 205. [Bay5b] T. Bayraktar. Global universality of zeros of random polynomials. preprint, 205. [BC3] T. Bayraktar and S. Cantat. Constraints on automorphism groups of higher dimensional manifolds. J. Math. Anal. Appl., (405):209 23, 203. [BBL96] E. Bogomolny, O. Bohigas, and P. Leboeuf. Quantum chaotic dynamics and random polynomials. J. Statist. Phys., 85(5-6): , 996. [BC2] C. Bordenave and D. Chafaï. Around the circular law. Probab. Surv., 9: 89, 202. [BD0] Jean-Yves Briend and Julien Duval. Deux caractérisations de la mesure d équilibre d un endomorphisme de P k (C). Publ. Math. Inst. Hautes Études Sci., (93):45 59, 200. [BK0] E. Bedford and K. Kim. Continuous families of rational surface automorphisms with positive entropy. Math. Ann., 348(3): , 200. [BL3] T. Bloom and N. Levenberg. Random polynomials and pluripotential-theoretic extremal functions. preprint arxiv: , 203. [Blo07] T. Bloom. Random polynomials and (pluri)potential theory. Ann. Polon. Math., 9(2-3):3 4, [Bro65] H. Brolin. Invariant sets under iteration of rational functions. Ark. Mat., 6:03 44 (965), 965. [BS07] T. Bloom and B. Shiffman. Zeros of random polynomials on C m. Math. Res. Lett., 4(3): , [Can99] S. Cantat. Dynamique des automorphismes des surfaces projectives complexes. C. R. Acad. Sci. Paris Sér. I Math., 328(0):90 906, 999. [Can0] S. Cantat. Dynamique des automorphismes des surfaces K3. Acta Math., 87(): 57, 200. [CLB05] S. Cantat and S. Le Borgne. Théorème limite central pour les endomorphismes holomorphes et les correspondances modulaires. Int. Math. Res. Not., (56): , [DGM3] C. D Andrea, A. Galligo, and S. Martin. Quantitative equidistribution for the solutions of systems of sparse polynomial equations. American Journal of Mathematics, 203. [DNS0] T.C. Dinh, V.A. Nguyên, and N. Sibony. Exponential estimates for plurisubharmonic functions. Journal of Differential Geometry, 84(3): , 200. [DT5] Henry De Thélin. Endomorphismes pseudo-aléatoires dans les espaces projectifs II. J. Geom. Anal., 25(): , 205. [Dup0] C. Dupont. Bernoulli coding map and almost sure invariance principle for endomorphisms of P k. Probab. Theory Related Fields, 46(3-4): , 200. [FLM83] A. Freire, A. Lopes, and R. Mañé. An invariant measure for rational maps. Bol. Soc. Brasil. Mat., 4():45 62, 983. [FS95] J. E. Fornaess and N. Sibony. Complex dynamics in higher dimension. II. In Modern methods in complex analysis (Princeton, NJ, 992), volume 37 of Ann. of Math. Stud., pages Princeton Univ. Press, Princeton, NJ, 995. [GZ3] S. Ghosh and O. Zeitouni. Large deviations for zeros of random polynomials with iid exponential coefficients. to appear in IMRN arxiv:32.695, 203.

9 RESEARCH STATEMENT 9 [Ham56] J. M. Hammersley. The zeros of a random polynomial. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, , vol. II, pages 89, Berkeley and Los Angeles, 956. University of California Press. [HN08] C. P. Hughes and A. Nikeghbali. The zeros of random polynomials cluster uniformly near the unit circle. Compos. Math., 44(3): , [HS95] B. Huber and B. Sturmfels. A polyhedral method for solving sparse polynomial systems. Math. Comp., 64(22):54 555, 995. [IZ3] I. Ibragimov and D. Zaporozhets. On distribution of zeros of random polynomials in complex plane. In Prokhorov and Contemporary Probability Theory, pages Springer, 203. [Kac43] M. Kac. On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc., 49:34 320, 943. [Lju83] M. Ju. Ljubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dynam. Systems, 3(3):35 385, 983. [LO43] J. E. Littlewood and A. C. Offord. On the number of real roots of a random algebraic equation. III. Rec. Math. [Mat. Sbornik] N.S., 2(54): , 943. [McM07] C. T. McMullen. Dynamics on blowups of the projective plane. Publ. Math. Inst. Hautes Études Sci., (05):49 89, [MR04] G. Malajovich and J. M. Rojas. High probability analysis of the condition number of sparse polynomial systems. Theoret. Comput. Sci., 35(2-3): , [Nag6] M. Nagata. On rational surfaces. II. Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 33:27 293, 960/96. [Roj96] J. M. Rojas. On the average number of real roots of certain random sparse polynomial systems. Lectures in Appl. Math., 32: , 996. [RS97] A. Russakovskii and B. Shiffman. Value distribution for sequences of rational mappings and complex dynamics. Indiana Univ. Math. J., 46(3): , 997. [Sib99] N. Sibony. Dynamique des applications rationnelles de P k. In Dynamique et géométrie complexes (Lyon, 997), volume 8 of Panor. Synthèses, pages ix x, xi xii, Soc. Math. France, Paris, 999. [ST04] M. Sodin and B. Tsirelson. Random complex zeroes. I. Asymptotic normality. Israel J. Math., 44:25 49, [SV95] L. A. Shepp and R. J. Vanderbei. The complex zeros of random polynomials. Trans. Amer. Math. Soc., 347(): , 995. [SZ99] B. Shiffman and S. Zelditch. Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys., 200(3):66 683, 999. [SZ03] B. Shiffman and S. Zelditch. Equilibrium distribution of zeros of random polynomials. Int. Math. Res. Not., ():25 49, [SZ04] B. Shiffman and S. Zelditch. Random polynomials with prescribed Newton polytope. J. Amer. Math. Soc., 7():49 08 (electronic), [SZ08] B. Shiffman and S. Zelditch. Number variance of random zeros on complex manifolds. Geometric and Functional Analysis, 8(4): , [SZ0] B. Shiffman and S. Zelditch. Number variance of random zeros on complex manifolds, II: smooth statistics. Pure Appl. Math. Q., 6(4, Special Issue: In honor of Joseph J. Kohn. Part 2):45 67, 200. [ZZ0] O. Zeitouni and S. Zelditch. Large deviations of empirical measures of zeros of random polynomials. Int. Math. Res. Not. IMRN, (20): , 200. Mathematics Department, Syracuse University 3244 NY, USA address: tbayrakt@syr.edu