STATISTICS ON HILBERT S SIXTEENTH PROBLEM

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1 STATISTICS ON HILBERT S SIXTEENTH PROBLEM ANTONIO LERARIO AND ERIK LUNDBERG Abstract. We study the statistics of the number of connected components and the voume of a random rea agebraic hypersurface in RP n defined by a homogeneous poynomia f of degree d in the rea Fubini-Study ensembe. We prove that for the expectation of the number of connected components: (1) Eb 0(Z RP n(f)) = Θ(d n ), the asymptotic being in d for n fixed. We do not restrict ourseves to the random homogeneous case and we consider more generay random poynomias beonging to a window of eigenspaces of the Lapacian on the sphere S n, proving that the same asymptotic hods. As for the voume properties we prove that: () EVo(Z RP n(f)) = Θ(d). Both equations (1) and () exhibit expectation of maxima order in ight of Minor s bound b 0(Z RP n(f)) O(d n ) and the bound Vo(Z RP n(f)) O(d). 1. Introduction Hibert s sixteenth probem, in its genera form, asks for the study of the maxima number and the possibe arrangements of the components of a nonsinguar rea agebraic hypersurface of degree d in RP n. It turns out that even the case of curves in RP is an extremey subte probem and essentiay nothing is known for n 3. The anaogous question for a hypersurface in CP n is trivia, as for a generic choice of the poynomia defining it, its topoogy is determined (this is the statement that the cohomoogy of a compete intersection in the compex projective space is determined by its mutidegree and compete intersections are generic). Thus, if from one hand no technique exists to study the arrangement of the components of a rea agebraic hypersurface, on the other hand one sti woud ike to formuate typica statements: a way for doing that is to repace the word generic with random. Let us start by considering the case of hypersurfaces in RP 1, i.e. zeros of ordinary univariate poynomias. In fact here we consider the homogenization of a univariate poynomia together with its projective roots rather than the cassica poynomia itsef, but it is cear that the two objects produce the same statistic. Over the compex numbers statements ike: a generic compex poynomia of degree d has d roots make perfect sense. On the other hand the ony property we can say to be generic for the rea roots of a rea poynomia is the one of being distinct. 1

2 ANTONIO LERARIO AND ERIK LUNDBERG Proposing a random viewpoint raises the obvious question: what is meant by random? We first ook to the semina work [14] of Kac, who proved that a random univariate poynomia of degree d whose coefficients are i.i.n. has asymptoticay /π og d rea roots. Even if it might seem natura, the distribution Kac considered is not invariant by the action of projective transformations over the extended rea ine. To produce such an invariant distribution we proceed as foows. We first extend our space of poynomias by homogenization; this eads us to consider projective zeros rather than just on the rea ine. Thus, we consider the vector space W 1,d of rea homogeneous poynomias of degree d in two variabes. We endow this vector space with a Eucidean structure by considering the L -norm of the restriction of these poynomias to the circe S 1. In an equivaent way we can define for f and g in W 1,d their scaar product to be: f, g = f(x)g(x)e x dx, R where x is the variabe in R. Notice that since f and g are homogeneous of the same degree, the previous integra equas the integra of fg restricted to the circe S 1 (up to a constant factor). Using this scaar product we can define a centered Gaussian probabiity distribution on W 1,d by etting for a measurabe subset A: probabiity of A = 1 e f df, ν 1,d where ν 1,d = (π) (d+1)/ is the constant that makes the previous a probabiity distribution and f is treated as the integration variabe that runs over W 1,d. For us a random poynomia wi simpy be a poynomia in W 1,d with the above distribution. Having thus repaced generic with random, we can make a positive statement in the rea setting: d(d + ) a rea random poynomia of degree d has roots. 3 We provide now another equivaent definition of a random poynomia. We fix a basis {f 1,..., f d+1 } for W 1,d which is orthonorma for the above scaar product and we consider a random function of the form: d+1 f = ξ i f i with ξ i N(0, 1). i=1 What we get in this way is again a random poynomia in the above sense. One way to produce such a basis is by considering trigonometric poynomias, i.e. functions on the circe satisfying: A S 1f = f, N. We denote by H 1, the vector space of a the soutions of the previous equation: these must be inear combinations of sin(θ) and cos(θ). Equivaenty we can extend such functions by homogeneity to a R and we get harmonic poynomias of degree. What

3 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 3 is interesting for us is that in this fashion we can write the vector space W 1,d as: W 1,d = x d H 1,, d N where again x is the variabe in R and the direct sum turns out to be orthogona with respect to the above scaar product. In particuar the above orthonorma basis can be obtained by coecting together the orthonorma bases coming from each H 1,, d N. If we repeat the same construction for the scaar product f, g C = f(z)g(z)e z dz, C then the corresponding rea random poynomias are caed Kostan distributed (or Bombieri-Wey). The vector space is the same, but in this second case the Eucidean structure comes by restricting to the rea poynomias the Hermitian structure given on the compex ones. The expectation of the number of rea roots of a rea Kostan poynomia of degree d is d. Aready from this comparison we see that the structure of the zeros of a random poynomia is richer than the one of a Kostan, which in turn is richer than for the naive ensembe. Remark 1. The two numbers d(d+)/3 and d are caed the parameters of these distributions. More generay each Gaussian distribution on W 1,d that is invariant by projective transformation has its parameter δ which is defined to be the square of the expected number of zeros it produces. Our interest is in what happens when we increase the number of variabes. We et W n,d be the vector space of homogeneous poynomias of degree d in n + 1 variabes, and consider their zero sets as hypersurfaces in RP n. If in the univariate case as a measure of our set we coud use the number of its points, here we have at east two possibe approaches to generaize this idea. The first one is by considering the number of points as a zero-dimensiona voume. In this case an upper bound for the size of our set is provided by 1 : (3) Vo(Z RP n(f)) Θ(d). Remark. The previous formua comes from an appication of the Integra Geometry formua and Wirtinger s inequaity (see [0]). In fact if f is reguar the previous voume is bounded by d Vo(RP n 1 )/Vo(CP n 1 ) and since d = Vo(Z CP n(f)) this aso gives a nice way to compare the rea zero ocus to the compex one (as we did in the univariate case using the Fundamenta Theorem of Agebra). The second approach is more subte and regards the number of points as a topoogica property (rather than a metric one), namey as the number of connected components. It is interesting to notice that there is a quantitative bound aso for this number, usuay referred to as Minor s bound (see [19]): (4) b 0 (Z RP n(f)) Θ(d n ). 1 Here the notation f(d) = Θ(g(d)) means that there exist two constants c1, c such that for d arge enough c 1g(d) f(d) c g(d).

4 4 ANTONIO LERARIO AND ERIK LUNDBERG Remark 3. The origina paper [19] of Minor does not contain the bound in this form, since a d n+1 term appears. We wi give a proof of (4) in the Appendix for the smooth case (which is the case we are interested in, since the set of nonreguar poynomias has measure zero in W n,d ), though the same bound hods true even for singuar hypersurfaces. As we aready pointed out, for a generic f in W n,d both the voume and the Betti numbers are constant for its compex zero ocus. This is certainy not the case for the rea zero ocus, as for exampe it might even be empty. To study the expectation of these quantities over the reas we make W n,d into a Gaussian space. We first define the scaar product of two eements f and g in it to be: f, g = f(x)g(x)e x dx. R n+1 Using this scaar product W n,d is turned into a probabiity space by defining for each of its measurabe subsets A: probabiity of A = 1 e f df, ν n,d A where now ν n,d = (π) dim(wn,d)/. Foowing [3], we ca W n,d with the resuting distribution of probabiity the rea Fubini-Study ensembe. The expectation of the voume of the zero ocus of a random poynomia of degree d from the rea Fubini-Study ensembe was computed by P. Bürgisser in [7]: d(d + ) (5) EVo(Z RP n(f)) = Vo(RP n 1 ). 3 In fact in the paper [7] this resut foows from the more striking computation of the expected curvature poynomia of Z RP n(f) for any projectivey invariant distribution. Such a resut is formuated in terms of the above mentioned parameter δ, which in this case is the square of the average number of rea zeros of a restriction to an RP 1. In particuar, if we define the Kostan distribution on W n,d by means of the scaar product C f(z)g(z)e n+1 z dz, the corresponding parameter is d and the resut we get for the expectation of the voume is EVo(Z RP n(f)) = dvo(rp n 1 ) (the genera resut for a distribution with parameter δ is δ 1/ Vo(RP n 1 )). In the case n = and restricting the distribution of probabiity to H,d, i.e. considering ony harmonic poynomias rather than homogeneous ones, the statistics of the number of connected components was studied by Nazarov and Sodin. In the breakthrough paper [] the authors prove that there exists a constant c > 0 such that for a random spherica harmonic f of degree d: Eb 0 (Z im S (f)) d d = c. Moreover, they showed that b 0 concentrates exponentiay about its mean. The corresponding norm is usuay caed the Bombieri norm.

5 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 5 In the visionary handwritten etter [3] P. Sarnak caims that using the same ideas as in [] it is possibe to extend this resut to the case of rea homogeneous poynomias in the rea Fubini-Study ensembe. The interesting part of these statements (besides the concentration) is the ower bound estimate: an upper bound of the same order is given by Minor s one (or in the spherica harmonic case by Courant s noda domain theorem). The statistics of the number of connected components of a Kostan distributed curve was studied by Gayet and Weschinger: in the paper [11] the authors prove that such a rea curve of degree d has on average O(d) components. The same authors prove in [1] that the expectation of each Betti number of a Kostan hypersurface in RP n is bounded by O(d n/ ), generaizing in some sense the univariate case. On a different perspective the first author in [16], using techniques from Random Matrix Theory, has studied intersection of one or two Kostan distributed quadrics, the asymptotic being this time in the number n of variabes; for such random agebraic sets the expectation of the sum of the Betti numbers behaves asymptoticay as n. As we did before, it is possibe to give an equivaent definition of the rea Fubini-Study structure on W n,d using an L (S n )-orthonorma basis. In fact we can again decompose our vector space into the orthogona sum: W n,d = x d H n,, d N where now H n, is the vector space of spherica harmonics of degree, i.e. those functions on the sphere S n satisfying: S nf = ( + n 1)f. Inspired by the ideas of [] and [3] we study a more genera cass of random poynomias that contains as specia cases both the homogeneous and the harmonic ones. To define such a cass, for every α [0, 1] and d > 0 et us consider the foowing index set: [αd, d] = {natura numbers such that [αd] d and d is even}. With this definition the α-window of eigenspaces we are going to consider is the foowing orthogona direct sum: H [αd,d] = H n,, α [0, 1]. [αd,d] We endow H [αd,d] with the probabiity distribution induced by the rea Fubini-Study ensembe. Remark 4. There is no unified convention for the names of these ensembes. In fact the name rea Fubini-Study shoud be distunguished from compex Fubini-Study which is another name for what we caed Kostan distributed. In this paper we wi ony dea with the rea Fubini-Study ensembe and its restriction to H [αd,d]. For this reason, since no confusion can arise, a random poynomia for us wi simpy mean a poynomia from this ensembe. In particuar for α = 0 we get random homogeneous poynomias of degree d and for α = 1 we get random spherica harmonics of degree d: H [0,d] = W n,d and H [d,d] = H n,d.

6 6 ANTONIO LERARIO AND ERIK LUNDBERG Remark 5. In fact we do not need to restrict to the case the α-window contains ony the eigenspaces H n, with d even. Our choice is motivated by the fact that we want to avoid redundancy in order to get random poynomias for α = 0; one can consider as we the fu window. We denote the dimension of H n, by d(n, ) and the one of H [αd,d] by D α ; notice that in the case α 1 we have D α = Θ(d n ). If we consider an orthonorma basis {Y i}d(n,) i=1 for H n,, we can decompose a random f in H [αd,d] into a sum of random spherica harmonics of different degrees: f = [αd,d] d(n,) i=1 ξ i Y i with ξ i N(0, D 1 α ). The introduction of the scaing coefficient Dα 1 is absoutey not necessary, but it has the advantage that E f L (S n ) = 1 which simpifies the notation in the proofs. As for the expectation of metric properties of the zero ocus of a random f in H [αd,d], we prove the foowing theorem. Theorem 1. Let α [0, 1] and f be a random poynomia in H [αd,d]. Then: EVo(Z S n(f)) = Θ(d). In particuar taking expectation on both sides of inequaity (3) turns it into an equaity (the voume on the projective space and the one on the sphere being reated by a constant that depends ony on n). We notice that the statement for α = 1 was proved by Bérard in [5] and for α = 0 by Bürgisser [7] as stated above in (5). In fact it is possibe to write the exact vaue of this expectation using an anaog of what we caed the parameter of the distribution, defined in this case as δ = 1 nd α [αd,d] ( + n 1)d(n, ); the expectation of the voume is then δ 1/ Vo(S n 1 ). The same statement hods for the projective zero ocus, as it is doube covered by the spherica one. The statistics of the number of connected components is more subte and requires more work, drawing on the ideas of []. The statement we wi prove is the foowing. Theorem. Let α [0, 1], and f be a random poynomia in H [αd,d]. Then: Eb 0 (Z S n(f)) = Θ(d n ). In anaogy with the case of the voume, this theorem says that if we take expectation on both sides of Minor s bound (4) we turn it into an equaity (the number of connected components on the projective space and those on the sphere being reated by a factor of two). Notice that the case n = and α = 1 gives the spherica harmonic case proved by Nazarov and Sodin in []. The case n = and α = 0 gives Sarnak s caim [3] on curves from the rea Fubini-Study ensembe. Remark 6. After this paper was written, the authors have been informed by F. Nazarov and M. Sodin that they have an aternative approach to this probem, that works in more genera settings; a summary of their ideas is avaiabe at [4].

7 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 7 The authors have aso been informed by P. Sarnak that together with I. Wigman they are abe to provide asymptotics for the expected number of ovas of a random pane curve for the naive mode (the defining coefficients for the monomias are independent normas with mean zero and variance one), the Kostan mode and the rea Fubini-Study one A random version of Hibert s sixteenth probem. It is interesting to discuss the case α = 0 and n =, i.e. random rea agebraic curves. The first part of Hibert s sixteenth probem asks to study the configuration of the components of a rea agebraic curve of degree d in the pane RP. The nuhomotopic components of such curves are caed ovas; each one of them separates RP in two components: one of them is homeomorphic to a disk (and is caed the interior of the ova) and the other to a Moebius band. In the case d is even a components are ovas, but if d is odd there is a component that is not an ova (it is homotopic to a projective ine and generates the fundamenta group of RP ). Harnack s theorem states that such curves cannot have more than (d 1)(d ) + 1 components, but the possibe arrangements of their ovas is a far more compicated probem; we refer the reader to the beautifu survey [6] on the subject. In fact the probem itsef of constructing curves with many ovas is a difficut one. Gayet and Weschinger s theorem [10] on the exponentia rarefaction of maxima curves confirms such difficuty: if we write the equation of a curve putting some coefficients in front of the standard monomias, it is exponentiay difficut to get one with many ovas. This approach of buiding curves depends on the choice of the basis we want to use for W,d ; if instead of the monomia basis (which is orthogona for the Bombieri scaar product, inducing the Kostan distribution) we use the spherica harmonics the resut improves dramaticay. In fact now, because of Theorem for α = 0 and n = the expectation of the number of components of the curve we obtain is a fraction of the maxima one. The price we pay is that we are no onger aowed to use the simpe monomia basis and we need an expicit expression for the poynomias representing spherica harmonics, such an expression being quite compicated. A competey anaogous discussion hods for rea hypersurfaces: now the fact that what we get appying Theorem is on average a fraction of the maxima number of components is due to Minor s bound. The method empoyed here (based on the ideas of Nazarov and Sodin) suggests that more generay one coud ask for the expected arrangement of such ovas. In fact essentiay the proof reies on the foowing observation: given any point x on the sphere and a sma disk around it of radius Θ(d 1 ) there is a positive probabiity (independent of d) of finding an ova inside such sma disk (and such ova oops around x). Unfortunatey this probabiity, even if constant, is sti very itte. A numerica computation of the expectation of the number of components for n = was performed by M. Nastasescu in [1]: as Sarnak writes in his etter a random curve is 4% Harnack. Thus, the random version of Hibert s Sixteenth Probem, to study the expected arrangement of a random curve, is sti open and despite its difficuty it seems to be very fascinating. One of the main difficuties in this framework is to define a statistic (beyond the number of components) that captures some features of the arrangement. We suggest one that we ca the energy of the curve C.

8 8 ANTONIO LERARIO AND ERIK LUNDBERG First we say that one ova C 1 is contained into another one C if it ies in the interior of C. Using containment we can give a partia ordering on ovas and construct a forest (a coection of trees) representing their arrangement. The vertices of the trees represent the ovas, with the empty ovas represented by eaves, and an ova not contained in any other is represented by a root. We define an energy that is additive for disjoint unions of trees, and mutipicative when appending trees. So the energy of a forest is the sum of the energies of its trees. We define the energy of a tree with ony one isoated vertex as, and of a arger tree inductivey as twice the sum of the energies of a subtrees that resut from removing its root. This description determines the energy of any forest (for exampe, see Figure 1). Figure 1. A hypothetica arrangement of ovas for C and the corresponding forest. The energy is h(c) = ( + ) + ( + ) = 36. By Bézout s Theorem the ength of each tree cannot be more than d and if such a tree exists then no other trees can appear; the corresponding curve is said to be maximay nested and its energy is h(c) = d/. On a somehow opposite direction one can consider Harnack curves, having h(c) 5d (for d arge enough). Such curves have the maxima number of components and satisfy some specia condition on the arrangement of their ovas; we refer the reader to [18] for the exact definition and to [15] for surprising connections with panar dimers. Thus, using this notion, one open probem that we can state on the expected arrangement is to describe the asymptotic behavior as d goes to infinity of the energy of a random curve. The pictures we get of a random curve suggest that the energy has the same order as a Harnack curve, so we conjecture that: (the ower bound foows from Theorem ). og Eh(C) im = d og d 1.. Pan of the paper. In Section, we speciaize to the case n = 1. This heps to iustrate the main techniques used in the paper and provides a proof for the vaue of the parameter δ used Section 6.. Theorem 1 is proved in Section3 and Theorem in Section 4. Section 5 is devoted to the proof of the main technica emmas for Theorem. In Section 6 we discuss some exampes (more on random curves and random surfaces). In the Appendix we provide the proof of Minor s bound, we review a construction of hyperspherica harmonics, as we as some of their properties.

9 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 9 Acknowedgements The authors are gratefu to S. Basu for his constant support and many usefu suggestions and A. Lancichinetti for stimuating discussions.. A toy mode proof: the univariate case To iustrate the main ideas, we first consider a random poynomia in the specia case n = 1. As mentioned in the introduction, in this case both the zero dimensiona voume and the number of connected components coincide. For simpicity of notation we wi denote their expectation by: E d = E{number of zeroes of a rea random univariate poynomia}. In view of the generaization to higher dimensions, we wi give two proofs that E d = Θ(d). The first one is based on techniques coming from integra geometry and wi produce an exact resut; this proof wi generaize to any number of variabes producing again an exact resut. The second is based on the barrier method introduced by Nazarov and Sodin; it wi produce ony an approximate resut but has the advantage that it generaizes to higher dimensions. We notice that in the case n = 1, a rea random poynomia is just a trigonometric poynomia with either ony even or ony odd terms (depending on whether d is even or odd)..1. Integra geometry proof. The first proof is obtained foowing the ideas of [9]. Reca that if we have functions {f 1,..., f k } defined on the rea ine we can consider the random function: k f = ξ i f i, with ξ i N(0, 1). i=1 Then if the coefficients are independenty distributed, the expected number of zeros of f on the interva [a, b] is given by the formua: (6) E{number of zeroes of f on [a, b]} = 1 π b a γ(t) dt, where γ is the projection of the curve c : t (f 1 (t),..., f k (t)) on the unit sphere S k 1 (we refer the reader to the beautifu paper [9] for this and reated topics). We note that (6) is proved using a specia case of the kinematic formua from integra geometry. In our case [a, b] = [0, π] and if d is even, d/ 1 ( sin(kt) cos(kt) f(t) = ξ 0 + a k + b k ), ξ 0, a k, b k N(0, 1). π π π k=1 The curve c is given by: c(t) = 1 ( ) 1, sin(t), cos(t),..., sin(td), cos(td) π

10 10 ANTONIO LERARIO AND ERIK LUNDBERG and its norm in R k is constant and equas π γ(t) = c(t) and d + 1 d+1 π. In particuar we have: π γ = ċ d + 1. The norm of γ is γ = (0, cos(t), sin(t),..., d cos(td), d sin(td)) d + 1, which easiy simpifies to: d(d + ) γ =. 3 A simiar cacuation in the case d is odd gives the exact same resut. We are now in the position to use formua (6) and we obtain that the expected number of zeros of f on d(d+) [0, π] is 3. Thus, the number E d of expected zeros on the rea projective ine is: d(d + ) E d =. 3 Notice that this is an exact resut and gives the asymptotic E d = Θ(d)... Barrier method proof. In this section we ony prove that E d cd for d arge enough; the inequaity E d d foows from the Fundamenta Theorem of Agebra. The proof is centered around the function described in the foowing Caim whose proof is deferred unti the end. Caim (Existence of the barrier). There exist numerica constants ρ and c 1 such that, for each sufficienty arge d and each x S 1 there is a homogeneous poynomia b x of degree d and L -norm one satisfying: b x (x) c 1 d and bx (e iρ/(d+1) x) c 1 d. Once a point x in S 1 is fixed, we can decompose f = ξ 0 b x + f x where ξ 0 is a Gaussian random variabe with variance 1 d+1, and f x is in the orthogona compement to b x in W 1,d. We choose a new Gaussian random variabe ξ 0 independent of ξ 0 and of f x with variance 1 d+1, and set: f ± = ± ξ 0 b x + f x. By construction the random functions f + and f have the same distribution as f. We notice that we can write: f = ξ 0 b x + 1 (f + + f ). Next we appy the foowing Caim to f + and f. Caim. There exists a constant c such that E f(x) c for every x in S 1. We aso defer the proof of this Caim in favor of first seeing how it is used. Let us reca that Markov s inequaity for a positive random variabe X reads P{ X c} c 1 E X.

11 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 11 If we appy such inequaity to X = f(x), because of the previous Caim we can find a constant c 0 such that: P{ f ± (x) c 0 } 1/5 and P{ f ± (e ir/(d+1) x) C 0 } 1/5, where ρ is provided by the Existence of the barrier Caim. The event that f ± (x) and f ± (e iρ/(d+1) x) are each simutaneousy at most c 0 is an intersection of four events each of which is compementary to the one of the above. Thus the probabiity of their intersection satisfies: P ( ) = 1 P ( ) 1 4 1/5 = 1/5. We note that these four events are not mutuay independent, but this is not needed in the ine above. Now consider the event Ω(x) that f(x) c 0 and f(e iρ/(d+1) x) c 0. If Ω(x) occurs then f has a zero 3 on S 1 between x and e iρ/(d+1) x. The event Ω(x) happens provided that: (i) ξ 0 d c 1 1 c 0 and (ii) f ± evauated at at x and e iρ/(d+1) x are each bounded in absoute vaue by c 0. Note that (i) and (ii) are independent, and we have just checked above that (ii) happens with probabiity at east 1/5. Moreover, P{ξ 0 d c 1 1 c 0} c 3, for some constant c 3, since ξ 0 d has variance d d+1 = Θ(1). It remains to choose Θ(d) disjoint intervas in S 1 each of ength ρ/d. Each of them contains a zero of f with probabiity at east c 3 /5 and hence: as desired. E d c d, First Caim. We wi choose our barrier in such a way that the barrier b x centered at some point x S 1 is obtained just by precomposing b (0,1) with a rotation that moves the point (0, 1) to x, so we ony need to define b (0,1) which we do by normaizing the function: U d (e iθ sin((d + 1)θ) ) = U d (cos θ, sin θ) =. sin θ Notice that U d is a poynomia of degree d, since the denominator divides the numerator. (U d is a projectivized Chebychev poynomia of the second kind). We have: U d (0, 1) = d, say by de Hopita s rue with θ 0, and for a d sufficienty arge, we aso have ( ) U d e i 3π (d+1) 1 d. Thus it suffices to show that the norm of U d is on the order of d as d, for then we wi obtain the desired properties for b (1,0) (the normaization of U d ). It is easy to show that U d = πd; thus the function b (0,1) (x) = 1 πd U d 3 In this one dimensiona case the existence of a zero is essentiay the intermediate-vaue property. In higher dimensions this event wi be generaized by considering the boundary of a disk centered at x; in the current situation we ony need to use one of the two boundary points of an interva centered at x.

12 1 ANTONIO LERARIO AND ERIK LUNDBERG has the desired properties with c 1 = 1 π and ρ = 3π (d+1). Second Caim. Let us choose a new orthonorma basis g 0, g 1,.., g d so that a basis eements except g 0 vanish at x. To see that it is possibe to do this, first notice that the subspace V of homogeneous poynomias of degree d vanishing at x has codimension one inside W 1,d. Thus, to get the desired orthonorma basis, we may first orthonormaize a basis for V, then take its orthogona compement V which consists of just one function. Once normaized, this function gives the basis eement g 0. Writing our random poynomia in terms of this basis, we have f(x) = ξ 0 g 0 (x) + ξ k g k (x) = ξ 0 g 0 (x). k Using again the standard basis f k, the fact that g 0 is a normaized trigonometric poynomia of degree d means that g 0 (x) = A k f k (x) A k A k d + 1 = d + 1, k k k where we have aso used the triange inequaity and the discrete Cauchy-Schwarz inequaity. Combining this with the above, f(x) d + 1 ξ 0, which impies that E f(x) is bounded by a constant (the Gaussian random variabe d + 1 ξ0 has variance of constant order, and therefore so has the expectation of its moduus). 3. Proof of Theorem 1 We generaize the proof given in [5] for the harmonic case to our α-window of eigenspaces. Here the main idea is to use a basis of the eigenfunctions of the Lapacian to define a kind of Veronese embedding, transforming the geometry of zeros of such functions on S n to the geometry of the intersection of the image of S n with hyperpanes. What is nice about using harmonic functions is that such a Veronese embedding turns out to be a diation, i.e. diates the Riemannian metric by a constant factor. In the anguage of integra geometry such embeddings are caed moment maps (generaizing the idea of the moment curve γ that we used in the univariate case in Section.1). Let us denote by d(n, ) the dimension of H n, as above (thus in the case n = we have d(, ) = + 1 and in the genera case d(n, ) = Θ( n 1 )). For each (n, ) et {Y i } d(n,) i=1 be an orthonorma basis for H n, and consider the map: Λ n, : S n R d(n,) whose components are (Y 1,..., Y d(n,) ). Such a map has the foowing properties: (i) it is an immersion and its image is a submanifod of the sphere in R d(n,) of radius R = (d(n, )/Vo(S n )) 1/ ; (ii) it is a diation of ratio λr /n where λ = ( + n 1) it is the eigenvaue for the eigenspace H n, of S n. In other words the ength of the image of

13 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 13 a unit vector in S n under the differentia of Λ n, is λ/nr. Property (i) foows from the addition theorem for spherica harmonics: Z (x, y) = d(n,) i=1 Y i (x)y i (y), x, y S n. Here Z denotes the (reproducing kerne) zona harmonic of degree, for which Z (x, x) =. Thus, evauating the addition formua at y = x we get: d(n,) Vo(S n ) d(n, ) Vo(S n ) = Λ n,(x), which shows the image is contained in the sphere. Property (ii) is proved in [6] and goes under the name of nice imbeddings property. For a given α [0, 1) we consider thus the map: Λ : S n R Dα = R d(n,) [αd,d] whose components are the Λ n, normaized by the factor (D α /Vo(S n )) 1/, i.e. ( Vo(S n ) ) 1/ Λ = (Λ αd,..., Λ,..., Λ d ), d even. D α With this choice of the normaization the addition theorem for spherica harmonics impies that the image of Λ is contained into the unit sphere in R Dα. We caim now that Λ is a diation of ratio δ, as defined in the Introduction. To prove this et us consider a unit vector v T S n and et us compute the square of the norm of its image under the differentia of Λ: dλ(v) = [αd,d] Vo(S n ) D α dλ n, (v) = [αd,d] λ(n, )d(n, ) nd α = δ, where in the second equaity we have used the fact that Λ n, is a diation of ratio λ(n, )R /n. We appy now the integra geometry formua (see [9]): 1 (7) Vo(S Dα 1 Vo({a} Λ(S n ))da = Vo(Λ(S n )) Vo(Sn 1 ) ) S Dα 1 Vo(S n ). Property (ii) above impies that the map Λ is a covering space (in fact either a doube cover, or a diffeomorphism); et us say it is k-sheeted. We caim now that the eft hand side of (7) equas exacty kδ (n 1)/ EVo(Z S n(f)). In fact if a = (a 1,..., a Dα ), then {a} Λ(S n ) equas the image under Λ of the noda ine of the function D α i=1 a iy i, where {Y i } Dα i=1 is the basis for H [αd,d] obtained by coecting together the eements from the various orthonorma basis of H n,. Since Λ is a covering, the voume of such image (which is n 1 dimensiona) equas kδ (n 1)/ times the origina voume; moreover since our Gaussian distribution is uniform on the unit sphere in H [αd,d], then the above integra computes expectation and the concusion foows.

14 14 ANTONIO LERARIO AND ERIK LUNDBERG As for the right hand side of (7) again the fact that Λ is a k-sheeted covering and a diation of ratio δ, impies Vo(Λ(S n )) = kδ n/ Vo(S n ). Pugging a this into (7) we finay get: EVo(Z S n(f)) = δ 1/ Vo(S n 1 ). 4. Proof of Theorem Our proof is deepy infuenced by the ideas and the exposition of []. In fact for α = 1 our caim is the direct generaization of the ower bound estimate in []; this case is a simpe extension of the proof in [] and we devote a remark to it at the end of the proof. Notice that the upper bound for our expectation is provided by Minor s bound (4), discussed in the Appendix, and we ony have to prove the ower bound. Thus we fix 4 for the rest of the proof an α [0, 1). To start with, for x a point on the sphere S n we consider the foowing event: (8) Ω(x, r) = {f(x) > 0 and f D(x,r) < 0}. If Ω(x, r) occurs, then {f = 0} has a component inside D(x, r) (and this component oops around x). If we prove that for r = Θ(d 1 ) there exists a constant a 1 > 0 independent of x and d such that P{Ω(x, r)} a 1, then we can cover the sphere S n with Θ(d n ) disjoint such disks and get the statement (each one of these disks contributing at east a 1 to the expectation in the statement). Specificay, the radius r = r(d, n) we are going to use from now on is given by: r = y n d + n 1, where y is the first point of minimum of the Besse function J 0 and y n for n > is the first point to the right of zero where the Besse function J n reaches a minimum (in this case J n aready has a minimum at zero, see [5]). The foowing Lemma provides the existence of the so caed barrier function; we defer its proof unti the next section. Lemma 3 (Existence of a barrier). For every α [0, 1) there exists a constant c 1 = c 1 (α) > 0 such that for every d > 0 and every x in S n there exists a homogeneous poynomia B x of degree d and L -norm one satisfying: Using B x we can decompose H [αd,d] foowing decomposition for f: B x (x) c 1 d n/ and B x D(x,r) c 1 d n/. = span{b x } span{b x }, thus getting the f = ξ 0 B x + f with ξ 0 N(0, D 1 α ). 4 In particuar every constant in this proof is aowed to depend on α, but not on d.

15 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 15 Figure. The behavior of the barrier function B x at the point x. We et now ξ 0 be a random variabe distributed as ξ 0 but independent of it and we define f ± = ± ξ 0 B x + f. Notice that both f and f ± have the same distribution. The introduction of these new random poynomias aows us to write: (9) f = ξ 0 B x + 1 (f + + f ) and to spit our probem into the study of the behavior of B x and f ± separatey. In fact the event Ω(x, r) happens provided that for some constant a > 0 the two foowing events both happen: 1) E(x, r) = {ξ 0 B x (x) a and ξ 0 B x D(x,r) a }; ) G(x, r) = { f ± (x) a and f ± D(x,r) a }. To check that E(x, r) G(x, r) impies Ω(x, r) we simpy substitute the inequaities defining E(x, r) and G(x, r) in (9) evauating respectivey at x and at D(x, r). Now by definition the two events E(x, r) and G(x, r) are independent and thus: P{E(x, r) G(x, r)} = P{E(x, r)}p{g(x, r)}. Because of this it suffices to show that the probabiity of each one of them is bounded from beow by a positive constant that does not depend on d and this wi provide a bound from beow for the probabiity of Ω(x, r) which is independent of d. The fact that the probabiity of E(x, r) is bounded from beow independenty on d immediatey foows from Lemma 3: since D α = Θ(d n ) and B x (x) = Θ(d n/ ) the random variabe ξ 0 B x (x) is Gaussian with mean zero and variance Θ(1); simiary for the boundary. Thus for a 3 sma enough the probabiity of it being bigger than a 3 is uniformy bounded from beow. It remains to study G(x, r); to this extent notice that this event is given by the intersection of the four events G 1, = { f ± (x) a } and G 3,4 = { f ± D(x,r) a }. For these we do not need independence as: P { } {( ) c } { } G i = 1 P G i = 1 P 1 P{G c i} i i i G c i i

16 16 ANTONIO LERARIO AND ERIK LUNDBERG and it is enough to prove that the probabiity of the compement of each one of them is sma. To bound the probabiity of G c 1 and Gc we argue as foows. First notice that since f ± are distributed as f it is enough to prove the corresponding bound for a random f. We consider in H [αd,d] the hyperpane V of those functions vanishing at x together with its orthogona compement V. Such orthogona compement must be spanned by a norm-one function F of the form: (10) F = F 0 + y Y with y 1, [αd,d] [αd,d] where Y H n, is the normaized zona harmonic centered at x (notice in particuar that this normaization produced the r.h.s. inequaity in the above ine). In fact in each space H n, the orthogona compement of the zona Y centered at x vanishes at this point, and we immediatey get (10). Such zona function is obtained by composing the function in equation (19) with a rotation that moves the north-poe of S n to x; moreover by setting θ = 0 in equation (19) we see that it has order Θ( (n 1)/ ) at x. Using the above orthogona decomposition H [αd,d] = V V, we can write the random f as ξ 0 F + F with F a random function in V. Thus we get: (11) E f(x) E ξ ( ) 1/ 0 y Y (x) E ξ 0 Y (x) a 3. [αd,d] [αd,d] We have used (10) and the discrete Cauchy-Schwarz inequaity in the second inequaity; for the ast inequaity we have used the fact that Y (x) = Θ( n 1 ), which impies ( ) 1/ s(d) = Y (x) = Θ(d n/ ). [αd,d] Since the variance of ξ has order Θ(d n ), when we scae it by s(d) we make its variance of constant order, which gives (11). The statement on the probabiity of { f(x) a } being smaer immediatey foows from Markov s inequaity. We expain now the argument for the remaining two cases. Again since f ± is distributed as f, the probabiity of each G c 3,4 is exacty the probabiity of the foowing event: { f D(x,r) a for a random f H [αd,d] }. Hence we consider the positive random variabe: X = max D(x,r) f. Bounding the expectation of X is more deicate and we devote to it a separate emma, whose proof is postponed to the next section. Lemma 4. For every α [0, 1] there exists a constant c = c (α) such that E max D(x,r) f c. Appying Markov s inequaity to X combined with the previous Lemma gives a bound for the probabiity of G c 3,4, concuding the proof.

17 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 17 Remark 7. This proof does not appy when α = 1, but in that case one can use a rather direct generaization to n+1 variabes of the proof given in []. Namey, their Caim 3.1 providing a bound on the expected max over a sma disk is sti true in more variabes and can be proved in the same way, and the barrier function can again be chosen to be the normaized zona harmonic composed with a rotation. 5. Proofs of Lemmas 5.1. Proof of Lemma 3. We work out the detais for n = first. We wi construct B x for x the north poe and define it for the other points by considering the composition with rotations (such rotations preserve a the properties of the statement). Set y = y (the first minimum point of the Besse function J 0 ) and et Y 0 be the normaized zona harmonic of degree centered at the North Noe. Using the Legendre poynomia P of degree we can write: Y 0 (θ, φ) = P (cos θ) + 1, where (θ, φ) are the standard poar coordinates on S. We reca from [5] the foowing asymptotic for P (Hib s asymptotic): P (cos θ) = ( ) 1 ( ) θ + 1 J0 θ sin θ + R (θ), where J 0 is the Besse function of the first kind and order zero. The error R (θ) is aways ess than θ 1/ O( 3 ). The function J 0 satisfies the foowing properties: J 0 (0) = 1, it is decreasing in [0, y] where y is a oca minimum with J 0 (y) < 0. We ca z (0, y) the first zero of J 0. Let µ be the maximum of the two vaues α and z/y; we define B x as: B x = [µd,d] Y 0 Card[µd, d]. Since each Y 0 has norm one and since different zonas are pairwise orthogona, we immediatey see that the L -norm of B x is one, as required from the statement. Let us now estimate the vaue of B x at x. Using the integra estimate for the numerator, which provides a bound of the form Θ(d 3/ ), and the fact that Card[µd, d] = Θ(d 1/ ), for some constant a 1 > 0 independent of d we have: [µd,d] + 1 B x (x) = a 1 d. Card[µd, d] Reca that the number r was defined such that d+1 r = y, i.e. we set: r = y d + 1. We estimate now the vaue of B x on the boundary of a disk of radius r. Notice that it suffices to show the bound for the numerator in the definition of B x has order Θ(d 3/ ),

18 18 ANTONIO LERARIO AND ERIK LUNDBERG since we aready know the denominator has order Θ(d 1/ ). Because of Hib s asymptotic we have: Θ(d 1/ )B x (r) = ( r + 1 sin r J 0 r) R (r) + 1. [µd,d] [µd,d] }{{}}{{} L(r) R(r) We estimate the error R(r) first. Notice that the summands R (r) + 1 are a bounded by r 1/ O( 1 ) so that: ( d ) ( ) og d R(r) r 1/ O 1 = r 1/ O(og d) = O. d =1 Next we bound L(r) as foows. First notice that since µ = max{α, z/y}, then for every in the sum J 0 ( +1 r) < 0. We fix now a γ (µ, 1) and notice that since each term is negative and J 0 is decreasing in (γy, y), then: L(r) ( ) r 1/ J 0 (γy) + 1. sin r [γd,d] Moreover using the expansion (r/ sin r) 1/ = 1 + Θ(d 1 ) we get: L(r) J 0 (γy)(1 + Θ(d 1 ))O(d 3/ ) c d 3/, where we have used the integra estimate for the sum of the square roots. Recaing that L(r) = B x (r)θ(d 1/ ), this concudes the proof for n =. For the case n > we argue as foows. First reca that we set y n to be the first point after zero where the Besse function J n reaches a minimum. The function J n has the foowing properties: it vanishes at zero and we denote by x n its first point of maximum (it is positive at this point); it is decreasing on the interva [x n, y n ] and negative at y n. We et z n be its ony zero in [x n, y n ] and µ n be max{α, z n /y n }. If Y H n, is the normaized zona harmonic centered at x, we define the barrier as: [µ B x = Y nd,d] Card[µn d, d]. Since the zonas are pairwise orthogona, the L -norm of B x is one. As aready discussed in the proof of Theorem, the order at x of each zona Y is Θ( (n 1)/ ) as shown in (18), and thus the order of their sum at x is Θ(d (n+1)/ ). Moreover since Card[µ n d, d] = Θ(d 1/ ), then we immediatey get that for some positive constant c 1 : B x (x) c 1 d n/. The proof of the bound for the vaue of B x on D(x, r) invoves the same steps as in the case n =, this time using a generaization of Hib s asymptotic invoving the Besse function J n. We need an expicit expression for Y first: Y (θ) = c(n, )P ( n, n ) (cos θ),

19 STATISTICS ON HILBERT S SIXTEENTH PROBLEM 19 where P ( n, n ) is a Jacobi poynomia and c(n, ) has order Θ( 1/ ), see (0) in the Appendix. Using the asymptotic provided in ([5], Theorem 8.1.1) we have: ( P ( n, n ) (cos θ) = sin θ cos θ ) n { ) θ 1/ } h(n, )( J n (Nθ) + R (θ), sin θ where N = +n 1, h(n, ) = Θ(1) is given by (1), and the error term R (θ) is aways ess than θ 1/ O( 3 ). Since by definition r = yn d+n 1 we have (sin r cos r ) n = Θ(d n ), and pugging a this into the definition of B x, we have that B x (r) equas: Θ(d 1 n {( r )Θ(d ) Θ( 1/ ) sin r [µ nd,d] ) 1/ J n (r( + n 1)/) + R (r) For the error term we get a bound of the form O(d n ). As for the principa term, we fix as above a γ n (µ n, 1) and notice that since each summand is negative and J n is decreasing in (γ n y n, y n ), then: B x (r) Θ(d n 3 ) Θ( 1/ )J n (γ n y n ) c 1 d n/ [γ nd,d] where in the ast inequaity we have used the integra estimate again. 5.. Proof of Lemma 4. Since the distribution is invariant by rotation, it is enough to prove the statement in the case the point x equas the north-poe (0, 0,.., 0, 1). We need an expicit expression for a rea orthonorma basis of H [αd,d]. To this end, we use the inductive construction of an orthonorma basis for H n,, the space of hyperspherica harmonics in n+1 variabes of degree, given in terms of orthonorma bases {Y j (φ)} j Im for each space H n 1,m with m = 0, 1,..,, where I m is an index set of size I m = dim H n 1,m = d(n 1, m). Namey, the basis eements as reviewed in the Appendix are: Y m +m) m (θ, φ) = N m (sin θ) m P ( n 1 (cos θ)y j (φ). In the previous formua P (λ) k is the Gegenbauer poynomia [5, Formua (4.7.1)], θ is the ange from the north-poe, the muti-ange φ S n 1 gives the remaining coordinates, and the normaization constant N m is stated expicity in (16) (see aso [, Chapter 3] for an exposition of this inductive construction). Using this basis we can write the random f in H [αd,d] as: f = [αd,d] m=0 = [αd,d] m=0 ξ j Y m (θ, φ) j I m ξ j N m (sin θ) m P ( n 1 m j I m +m) (cos θ)y j (φ), }.

20 0 ANTONIO LERARIO AND ERIK LUNDBERG with each ξ j distributed as N(0, D 1 α ), with Dα 1 = Θ(d n ). In particuar, the restriction of f to D(x, r) is given by: d { f(r, φ) = ξ j N m P ( n 1 +m) } m (cos r) (sin r) m Y j (φ), m=0 L m j I m where the index set L m := { [αd, d] m} arises from having reversed the order of summation. Notice that the term in the arge brackets is a Gaussian random variabe itsef: it is a inear combination of independent random variabes, each one of mean zero and variance Dα 1. We denote such a random variabe by ˆξ j ; in other words we define: (1) ˆξj = ξ j N m P ( n 1 +m) m (cos r), j I m. L m By the addition formua for independent Gaussians we get that ˆξ j is distributed as a Gaussian with mean zero and variance σ(m, d) where: d (13) σ(m, d) { N m P ( n 1 +m) m (cos r) }. =m Remark 8. Most importanty, for each m the Gaussians ˆξ j are identicay distributed across a j within the index set L m. This foows from the fact that in the sum (1) the coefficient of ξ j depends on m but not j. Using this new notation we can rewrite: d f(r, φ) = ˆξj (sin r) m Y j (φ) = m=0 j I m d (sin r) m m=0 j I m ˆξj Y j (φ). Using the triange inequaity, we bound the expectation of X = max D(x,r) f as: d E X (sin r) m E max ˆξj Y j (φ) φ S m=0 n 1 j I m. By the previous remark the function: F (φ) = j I m ˆξj Y j (φ) is a random spherica harmonic on S n 1 with independent identicay distributed Gaussian coefficients. This aows us to appy the foowing Caim, whose proof is provided in the Appendix. Caim (A basic estimate for random spherica harmonics:). E max F (φ) σ(m, d) d(n 1, m), φ Sn 1 where reca that d(n 1, m) is the dimension of the space of spherica harmonics of degree m in n variabes.

21 Returning to E X, we have: (14) E X STATISTICS ON HILBERT S SIXTEENTH PROBLEM 1 d (sin r) m σ(m, d) d(n 1, m). m=0 It remains to estimate σ(m, d). We note that the Gegenbauer poynomias satisfy [5, Formua (7.33.1)]: max x [ 1,1] P ( n 1 +m) ( ) m (x) + m + n =. m Using this to estimate (13) we get: σ(m, d) 1 D α d ( ) + m + n (N m ). m =m Inspecting the constant (N m ) (see formua (16)), we notice that we can bound part of it by a constant: Γ( n+m+1 ) πγ( n+m )(n + m 1) a 0, (in fact this number goes to zero as m goes to infinity); in particuar we bound each term in the above sum by a constant times ( ) (n + m )!( + n 1)( m)! + m + n, ( + m + n )! m which simpifies to This finay gives: σ(m, d) a 1 D α ( ) + m + n ( + n 1). m d ( ) + m + n ( + n 1) m =m a 1 D α (m + n )! d ( + m + n ) m+n 1 =m a 1 (d + m + n )m+n D α (m + n )! a (d) m (m!) where a is a constant that does not depend on d. (d) In particuar σ(m, d) a m m!. Recaing that r = y d+n 1 y d (we have set y n = y for simpicity of notation) and using (sin r) m r m ( y m, d) we have in (14): d ( y m d (y) E X σ(m, d) d(n 1, m) a d) m d(n 1, m). m! m=0 m=0

22 ANTONIO LERARIO AND ERIK LUNDBERG We estimate d(n 1, m) a 3 m n a m 4, so that using the definition of the exponentia function we obtain: d (ya 4 ) m E X a 5 a 5 e ya 4 c. m! This concudes the proof. m=0 6. Exampes 6.1. More on the arrangements of random curves. Besides the number of ovas of a curve in the projective pane, there are other interesting invariants associated to it that are worth studying (e.g. the energy we defined in the Introduction). One cassica invariant is the number of empty ovas; et us denote it by ν 0. If the curve C has even degree d = k, then Arnod s inequaities (see [1] section 6) impy: ν 0 (C) b 0 (C) (k 1)(k ). In particuar if a curve has the maxima number of ovas, then ν 0 (C) k. The numerica resuts obtained by M. Nastasescu in [1] have shown that the constant in the bound for Eb 0 is very sma: Eb 0 (C) 1 5 d, and combining this with Arnod s bound woud ony give Eν 0 (C) 0. On the other hand, as stated in the Introduction, the barrier method can be used to give information aso on the expected arrangement of the components of the random hypersurface. In fact the event Ω(x, r) as defined above (8) impies the existence of an empty ova inside the disk D(x, r). Thus, pacing on the sphere Θ(d ) disjoint such disks gives the foowing: Average number of empty ovas: Eν 0 (C) = Θ(d ). Remark 9. We notice that every f W n,k defines a function on the rea projective space RP n and for a generic f the gradient of this function does not vanish on its zero ocus. In particuar if C is a component of Z RP n(f), then RP n \C has two components; if q : S n RP n is the quotient map, we define the interior of C to be the component of RP n \C whose preimage under q is not connected. As we did for curves, we say that a component C 1 of Z RP n(f) contains another C if C is contained in the interior of C 1 ; an empty component is one whose interior is connected. Using this notation the above coroary can be stated for hypersurfaces as we and provides the existence of Θ(d n ) empty components. Simiary we can define the energy of an hypersurface as we did for curves: additive on disjoint unions and mutipicative on nestings. Instead of aways taking the vaue, the seed we take for the energy of an empty component is the sum of its Betti numbers (in the case of an empty ova, this equas ). As for higher moments of b 0, Nazarov ad Sodin have proved that for α = 1 (random spherica harmonics) it exponentiay concentrates about its mean and Sarnak has informed the authors that the same resut hods for α = 0 (homogeneous poynomias). Thus the typica curve has Θ(d ) components. The authors expect the same resuts to hod for any n as we.