ON THE KAC-RICE FORMULA

Transcription

1 ON THE KAC-RCE FORMULA LVU. NCOLAESCU Abstract. This is an informal introduction to the Kac-Rice formula and some of its mostly one-dimensional applications. Contents. Gaussian random functions 2. The one-dimensional Kac-Rice formula 4 3. Some applications 9 4. Energy landscapes 5 Appendix A. A probabilistic dictionary 7 References 9. Gaussian random functions Let be an interval of the real axis. Fix smooth functions f, f,..., f N : R, and independent normal random variables X k, k =,,..., N, of mean and variance v k >, defined on the same probability space Ω, A, P. Form the linear combination F t = F ω t = X k ωf k t.. This is an example of random function. The value of this random function at each point t is a Gaussian random variable. As probability space Ω we can take Ω = R N+ equiped with the Gaussian measure N e ω 2 k 2v k P dω dω N = dω dω N. 2πvk The random variables X k are then the coordinate functions X k ω,..., ω N = ω k, k =,..., N. The covariance kernel of this random function is the function K : R defined by Ks, t = E F s F t = f j sf k tex j X k. j, Date: Started September 6,24. Completed on October 9, 24. Last revision October, 24. Notes for a Graduate Seminar talk, October 24.

2 2 LVU. NCOLAESCU Since EX 2 k = v k and the variables X j, X k, j k, are independent we have We deduce EX j X k = EX j EX k =, j k. Ks, t = v k f k sf k t..2 We denote by ẐF, the expected number of zeros of F on. Example. Random polynomials. Fix N independent normal random variables X,..., X N, with mean and varx k = v k, k. Then F t = X + X t k + + X N t N. is a polynomial of degree N whose coefficients are independent random normal variables. Observe that F t = k X k t k. The random variables k X k are independent normal variables with mean and variances v k. Thus the random polynomials F t and F t = F t have the same statistics and we deduce Ẑ F, [, = Ẑ F,, ]. Suppose additionally that the variances v k satisfy the symmetry condition n this case the random polynomials F t and v k = v N k, k =,,..., N..3 F t = t N F t = have the same statistics and we deduce that n X N k t k Ẑ F,, = Ẑ F,,. We deduce that if F satisfies the symmetry condition.3, then Ẑ F, R = 4Ẑ F,, = 4Ẑ F,,..4 Here are two famous examples of random polynomials satisfying the symmetry condition.3. The Kac statistics. n this case the random variables have the same variance v = v = = v N =. We deduce that in this case the covariance kernel is Ks, t = st k = stn+..5 st The Kostlan statistics. n this case the variances are N v k =, k =,,..., n. k

3 n this case the covariance kernel is ON THE KAC-RCE FORMULA 3 Ks, t = N st k = + st N..6 k The above construction can be generalized as follows. Consider a family of polynomials p t, p t,..., p N t, deg p k t = k, k =,, 2,... N. Fix N independent normal random variables X,..., X N, varx k = v k, k. Then random polynomial of degree N. F t = X p t + X p t + + X N p N t The Legendre statistics. Recall that the Legendre polynomials are obtained from the sequence of monomials t k k by applying the Gramm-Schmidt procedure with respect to the inner product in L 2 [, ], dt. Concretely, the degree n Legendre polynomial is p n t = n + 2 l nt, l n t = d n 2 n n! dt n t2 n..7 We can construct a random polynomial F N t = X k p k t, where X k are independent standard normal random variables, X k N, k. Using the Christoffel-Darboux theorem [22] we deduce that its covariance kernel is given by K N s, t = N + 2 l N+ tl N s l N+ sl N t. t s Example.2 Random trigonometric polynomials. We assume = [, 2π], N is even, N = 2m and f =, f 2k t = sinkt, f k t = coskt. Asume v 2k = v 2k = 2r k >. For uniformity we set r = v. n this case we have m Ks, t = r + 2r cosks coskt + sinks sinkt = r + 2 k= m r k cos kt s = r + k= m r k e iks t + e ikt s. k= n the special case when r = r = = r m = we deduce m Ks, t = + 2 cos kt s = k= = e i 2 2m+t s e i 2 2m+t s e i 2 t s e i 2 t s = sin m k= m e ikt s 2m+t s 2 sin t s

4 4 LVU. NCOLAESCU For any interval and any function F : R we denote by ZF, the number of zeros of f. When F is a random function ZF, is a random variable and we would like to investigate some of its statistical invariants. The simplest such invariants is its expectation. The Kac-Rice formula gives a description of the expected number of zeros of the random function F in the interval. 2. The one-dimensional Kac-Rice formula Consider the random function defined in the previous section F : R, F t = X k f k t. We would like to compute the expected number of zeros of F in. The key to this computation is Kac s counting formula. For any ε > define see Figure { η ε : R R, η ε y = 2ε, y < ε,, y ε. Figure. Approximating the Dirac function. Let us point out that the family η ε converges as ε to Dirac s δ-function, i.e., for any continuous function f : R R we have η ε tft = f. lim ε For every C -function F : R we set Z ε F, := R η ε F t F t dt. Definition 2.. a Fix a compact interval [a, b]. We say that a C -function f : [a, b] R is convenient if the following hold. fa fb. All the zeros of f are nondegenerate, i.e., if ft =, then f t. b We say that a C -function F : R R is convenient if it is proper and all its zeros are nondegernerate.

5 ON THE KAC-RCE FORMULA 5 Lemma 2.2 Kac s counting formula. a Suppose that the C -function F : [a, b] R is convenient. Then ZF, [a, b] = lim ε Z ε F, [a, b]. b Suppose that the C -function F : R R is convenient. Then ZF, R = lim ε Z ε F, R. c Let = [a, b] or = R and suppose that F : R is a C convenient function with ν zeros and κ < critical points. Then for any ε > we have Z ε F, ν + 2κ. 2. Proof. a Since F is convenient, it has finitely many zeros τ < < τ ν. The set C of critical points of F is compact and disjoint from the zero set of F because F is convenient. Thus ε := min F x >. x C Fix ε, ε. Then b η ε F t F t dt = F t dt. 2ε a { F <ε} The connected components of the open set { F < ε} are open intervals c, d [a, b] disjoint from the critical set and such that F c = F d = ε. On such interval F t has constant sign so that F d F c = d c F t dt = 2ε. This proves that each connected component of { F < ε} contains a unique zero of F. Obviously, any zero of F is contained in one such component. We reduce that the set { F < ε} is a disjoint union of open intervals c i, d i, i =,..., ν such that Thus, ε, ε we have b a η ε F t F t dt = 2ε τ i c i, d i, i. ν i= di c i F t dt = ν = ZF, [a, b]. b Since F is proper there exists a compact interval [ a, a] such that F t > for t > a. The restriction of F to [ a, a] is convenient and we have ZF, [ a, a] = ZF, R. Next observe that for ε, we have η ε F t F t dt = Hence lim ε R R a a η ε F t F t dt. a η ε F t F t dt = lim η ε F t = F t dt = ZF, [ a, a] = ZF, R. ε a c Since F has only finitely many critical points we deduce from Rolle s theorem that for any c R the equation F t = c has only finitely many solutions.

6 6 LVU. NCOLAESCU Fix ε >.The connected components of the set { F < ε} are bounded intervals a, b such that F a = F b = ε. Since the equation F t = ε has only finitely many solutions we deduce that { F < ε} has only finitely many components We have J l = a l, b l, l =,..., L. η ε F t F t dt = 2ε L l= J l F t dt. Denote by k l the number of turning points of F t in J l, i.e., the number of local points where F t changes sign. Let observe that if J l contains no turning points, then F is either increasing or decreasing on this interval so F a l F b l < and thus J l contains a unique zero of F. n particular, if J l contains no turning point, then L F t dt = 2ε. J l We set The above discussion shows that and thus 2ε l= L := { l =,..., l; J l contains no turning point }, L l= L := { l =,..., l; J l contains turning points }. #L = ν, #L κ, F t dt = F t dt + J l 2ε l L J l 2ε = ν + F t dt. 2ε J l l L l L Let now l L and supose that the turning points of F in J l are Then t < < t kl. J l F t dt F t dt = F a l F t + F t F t 2 + F t kl F b l 2εk l +. J l Hence F t dt 2k l + κ + #L 2κ. 2ε J l l L l L Let us apply Kac s counting formula to the random function in. F = F ω : R, F ω t = X k ωf k t, X k : Ω, A, P R. Here we assume that is either a compact interval or = R. We make the following assumptions about the random function. The random function F is almost surely convenient. A

7 ON THE KAC-RCE FORMULA 7 C > such that, almost surely, ZF, + ZF, < C. A 2 Let us point out that the random polynomials in Example. satisfy these assumptions. As for the random trigonometric polynomials in Example.2, assumption A follows from a direct application of Sard s theorem, while assumption A 2 follows by observing that a trigonometric polynomial of degree m has at most 2m zeros and at most 2m critical point. Let F ω be convenient. Then Kac s counting formula implies. Thus ẐF, = E ZF, = ZF ω, = lim ε Z ε F ω, Ω ZF ω, P dω = Ω lim Z εf ω, P dω. ε At this point we want to switch the order of operations Ω and lim ε. Using assumption A 2, Lemma 2.2c and Lebesque s dominated convergence theorem we deduce that lim Z εf, P dω = lim Z ε F ω, P dω Ω ε ε Ω = lim E η ε F t F t dt = lim E η ε F t F t dt. ε ε Hence ẐF, = lim ε E η ε F t F t dt. 2.2 To compute the expectation of η ε F t F t we observe that for each t the random vector Ω ω F t, F t R 2 is Gaussian and its covariance form is described by the symmetric matrix [ ] at b C t = t, b t c t where a t = E F t 2, b t = E F tf t, c t = E F t 2. We can describe the entries of C t in terms of the covariance kernel Ks, t = EF sf t. More precisely, we have We set a t = Kt, t, b t = K ts, t s=t, c t = K sts, t s=t. 2.3 t := det C t = a t c t b 2 t, v t := t a t, and we make another assumption on F t, namely t >, t. A 3 This assumption is automatically satisfied for the random functions in Examples. and.2 provided that N is sufficiently large. The Gaussian measure Γ C on R 2 with covariance form C is then Γ Ct dxdy = 2π e c 2 t tx 2 2 tbxy+a ty 2 dxdy. t We need N 2 in Example. and m 2 in Example.2.

8 8 LVU. NCOLAESCU We then have We have Thus 2π t We have E η ε F t F t = y Γ Ct dxdy 2ε x <ε = ε 2ε 2π y e c 2 t tx 2 2b txy+a ty 2 dy dx. t R ε c t x 2 2 t bxy + a t y 2 = 2 t 2 t R = 2 t a t y b tx a t 2 + t a t x 2 a t y 2 2b t xy + b2 t y e 2 t c tx 2 2b txy+a ty 2 dy = 2πat e x2 2a t We would like to conclude that ẐF, = lim E ε x 2 + a t c t b2 t a t = a t y b 2 tx + 2 t a t 2π t a t = e x2 y γ vt,btx/a t dy 2πat R }{{} Φ tx 2a t E η ε F t F t = ε Φ t xdx. 2ε ε R 2a t x 2. x 2 y e a t y b t x 2 2 t a t dy. lim E η ε F t F t = Φ t = y γ vt dy. 2.4 ε 2πat R η ε F t F t dt = lim E η ε F t F t dt = Φ t dt. ε For any random To do this we will invoke Lebesgue s dominated convergence theorem. variable X we have the inequality E X EX 2 = varx + EX 2 and we deduce y γ vt,btx/a t dy v t + b 2 t x2 /a 2 t v t + b tx. R a t Now observe that Φ t x is positive and for any x we have Φ t x vt + b t = t + b t 2πat a t 2π a t a 2 =: µt. t We now add another requirement on our random function F t. µtdt <. A 4 This requirement is automatically satisfied for the random trigonometric polynomials in Example.2 because in this case is compact. The requirement A 4 is also satisfied for the random polynomials in Example. because in this case a t is a polynomial of degree 2N, b t is a polynomial of degree 2N 2, c t is a

9 ON THE KAC-RCE FORMULA 9 polynomial of degree 2N 2 and t is an even polynomial of degree 4N 3. n particular deg t 4N 4 and we deduce t + b t a t a 2 = Ot 2 as t. t We deduce E η ε F t F t = ε Φ t xdx µt. 2ε ε nvoking 2.2, 2.4, A 4 and the dominated convergence theorem we conclude that ẐF, = lim ε Observe that Observe that E η ε F t F t dt = Φ t = 2 γ vt ydy = 2πat π a t v t ρ 2 t = a tc t b 2 t a 2 t lim E η ε F t F t ε = Ks, t s=tk sts, t s=t K ts, t 2 s=t Ks, t 2 s=t We have thus proved the Kac-Rice formula. dt = ye y 2 2v t dy = π ρ t t, ρ t :=. a t = 2 st log Ks, t s=t. Φ t. Theorem 2.3. Let = [a, b] or = R. Suppose that f, f,..., f N : R are smooth functions and X k Nv k, k =,..., N are independent normal random variables. We form the random function with covariance kernel F : R, F t = X k f k t Ks, t = E F sf t. f the random function F t satisfies the assumptions A, A 2, A 3, A 4, then the expected number of zeros of F in is ẐF, = ρ t dt, ρ t = st 2 π log Ks, t Ks, tk st s=t = s, t K s, t 2 s=t Ks, t Remark 2.4. The quantity π ρ t in 2.5 is the expected density of zeros of F at t, i.e., π ρ tdt is the expected number of zeros in the infinitesimal interval [t, t + dt]. 3. Some applications We describe a few immediate consequences of Theorem 2.3. Example 3. The Kac statistics. Suppose that F N t is the degree N random Kac polynomial F N t = X k t k, 3.

10 LVU. NCOLAESCU where the random variables X are independent normal variables X k N. n this case the covariance kernel is K N s, t = stn+. st Then We deduce that 2 st log K N s, t s=t = t 2 2 N + 2 t 2N t 2N+2 2 =: f Ntdt. Z N := ẐF N, R = 4Ẑ F N, [, = 4 π Theorem 3.2 Kac, Edelman-Kostlan. As N we have fn tdt. Z N = 2 π log N + C + o, 3.2 where C = log 2 + x 2 sinh 2 x dx x + Proof. We follow the approach [7, 2.5]. We make the change in variables t = + x N and we deduce Z N = 2 gn xdx, π where 2 g N x = x + x/2n }{{} =A N x 2 2 N+ N+ 2 N + x/n + x/n 2N } {{ } =:B N x 2 The function g N has an apparent pole at x =, but it has an extension as a smooth function on [,. Note that The equality implies that The function lim A Nx = A x :=, x >. 3.3 N x lim B Nx = B x := N lim + N x/nn = e x 3.4 2ex e 2x =, x >. 3.5 sinh x g x = A x 2 B x 2 = x 2 sinh 2 x = sinh2 x x 2 x 2 sinh 2 x, x > extends by continuity at x = because. Moreover sinh 2 x x 2 = x4 3 + Ox6 as x. lim g Nx = g x uniformly for x [, ]. N

11 ON THE KAC-RCE FORMULA We have gn x A N x = A n x 2 B N x 2 B N x A N x = gn x + A N x lim N B N x gn x + A N x = B x g x + A x. The functions g N x A N x are integrable on [, and we have 2 lim N = We deduce that as N we have gn xdx = A N x + Now observe that and gn x A N x = B x g x + A x dx g x A x dx = g x dx. x x + dx + A N x = g xdx + g x dx + o. x x dx = log 2 + log 2 =, x + x dx = log2n +. 2N + x We deduce g gn xdx = log2n + + x x + g = log N + log 2 + x x + dx + o. dx + o Remark 3.3. a The graph of ρ n t = t 2 2 N + 2 t 2N t 2N+2 2 is depicted in Figure 2. t has two peaks at t = ± which suggest that the real roots of a Kac random polynomial tend to concentrate near t = ±. This statement can be made much more precise. n [, 2, Lemma ] it is shown that for any s, ] the expected number of roots in the interval + log N s, log N s is Olog N s log log N as N, Ẑ F Kac, + log N s, log N s = Olog N s log log N. 2 The passage to limit can be justified by invoking the dominated convergence theorem whose applicability can be verified upon a closer inspection of 3.3, 3.4 and 3.5.

12 2 LVU. NCOLAESCU Figure 2. The density of zeros of a random Kac polynomial of degree 25. b The asymptotics of the variance V N of the number of roots of a Kac random polynomial of large degree N was described by N.B. Maslova, [3]. More precisely, she proved that V N 2 log N as N. 4π π Using Chebyshev s inequality we deduce that for any k >, the probability that a random Kac polynomial has more than 2 π + k log N real roots is smaller than k 2 as N. log N c ntuitively, one should expect that a random polynomial has, on average many more critical points than zeros. M. Das [6] has shown that the expected number of critical points of a random Kac polynomial of degree N is asymptotic to + 3 π log N. This confirms the intuition, but it also shows that expected number of critical points has the same order of growth as the expected number of zeros. d A weaker version of the asymptotic estimate 3.2 is valid for more general classes of random polynomials. More precisely bragimov and Maslova have shown in [] that if F n t is a random degree N polynomial of the form F N t = X k t k, where X k k are independent identically distributed L 2 -random variables, then ẐF N, R 2 log N as N. 3.6 π The proof in [] is much more complicated and is based on ideas developed by Erdös-Offord [8] where they discuss the special case when X k are Bernoulli variables taking values ± with equal probability. Recently 24 H. Nguyen, O. Nguyen and V. Vu proved in [6] that if X k L p for some p > 2, then 3.6 can be refined to ẐF N, R = 2 log N + O as N. 3.7 π

13 ON THE KAC-RCE FORMULA 3 e Any polynomial of degree n has n complex roots. A degree N random Kac polynomial 3. F N t = F N ω, t has with probability N distinct complex roots. As n, these roots tend to distribute uniformly in annuli around the unit circle z =. More precisely, if α, β [, 2π, α < β and δ,, and we denote by Z N ω α, β, δ the expected number of zeros of F N ω, t in the region α < arg z < β, δ < z < + δ, then the random variable ω N Z Nω α, β, δ converges almost surely and in any L p, p, to β α 2π. For proofs and details we refer to [2], [4, 8.2], [2]. Example 3.4 The Kostlan statistics. Consider the Kostlan random polynomials P N t = X k t k where X k are independent normal random variables with zero means and variances N varx k =. k n this case the covariance kernel is We have The density of zeros is then Ks, t = + st N. log Ks, t = N log + st, t log Ks, t = st 2 + st st log Ks, t = N + st 2 = 2 st log Ks, t s=t = N + t 2. Ns + st, N + st 2. The Kac-Rice formula implies that the expected number of zeros is 2 N π + t 2 = N. We see that the Kostlan random polynomials have, on average, more real zeros than the Kac random polynomials. One can show that the expected number of critical points of a Kostlan random polynomial is equal to 2 3N 2; see [7]. Remark 3.5. Consider the complex analog of the Kostlan random polynomial F N ω, z = X k ωz k,

14 4 LVU. NCOLAESCU where X k : Ω C are independent complex valued random variable normally distributed with mean zero and variances N var X k =. k The distribution of zeros of F N ω, z in the complex plane C has many beautiful properties, [, 2]. For any Borel set B C we denote by vol S 2B the area of B viewed as a subset of S 2 via the stereographic projection from the North Pole to the plane that cuts the Equator. f Z N B denotes the expected number of critical points of F N in B, then see [, 2] Z N B = N 4π vol S 2B. n other words, we expect the zeros of the random polynomial F N ω, z to be uniformly distributed with respect to the probability measure 4π vol S 2 on C. As explained in [2], this happens with high probability as N. Example 3.6 The Legendre statistics. Consider the random polynomial F N t = X k p k t, where X k are independent standard normal random variables, and p k is the degree k Legendre polynomial defined in.7. M. Das [5] has shown that the expected number of zeros of F N t in [, ] is asymptotic to 3 N for large N. Example 3.7 Fáry-Milnor. Suppose that [, L] s rs = xs, ys, zs R 3 is an immersed curve smooth parametrized by arclength. We obtain a random function on the curve F s = Axs + Bys + Czs where A, B, C are independent standard normal random variables with mean zero and variance. ts derivative F s = Ax s + By s + Cz s has covariance kernel Ks, s 2 = x s x s 2 + y s y s 2 + z s z s 2 = T s T s 2, where T, N is the Frenet frame along the curve and denotes the standard inner product in R 3. We have s2 Ks, s 2 = T s T s 2 = κs 2 T s Ns 2, where κ denotes the curvature of the curve. Similarly 2 s s 2 Ks s 2 = κs κs 2 Ns Ns 2. We deduce Ks, s =, s2 Ks, s =, s 2 s 2 Ks, s = κs 2. The Kac-Rice formula implies that the expected number of critical points of F s along the curve is L κsds = the total curvature of the curve. π π

15 ON THE KAC-RCE FORMULA 5 This result was first independently and by different methods by. Fáry [9] and J. Milnor [4]. Example 3.8 Trigonometric polynomials. Consider random trigonometric polynomials of the form F m t = A + m 2 A k cos kt + B k sin kt, t [, 2π] where A, A,..., A m, B,..., B m are independent standard normal random variables. The covariance kernel of this random function is described in.9, m Ks, t = + cos kt s. We have We deduce k= m m t Ks, t = 2 k sin kt s, stks, 2 t = 2 k 2 cos kt s. Kt, t = 2m =, 2 stkt, t = 2 m k 2 = mm + 2m +. 3 We conclude that the density of zeros is ρ m t = mm +, π 3 so that the expected number of zeros in [, 2π] of this random trigonometric polynomial is mm + 2 2m as m Energy landscapes n this last part will discuss possible generalizations of the example involving trigonometric polynomials. We start with a few simple observations. A 2π-periodic function fθ can be viewed as a function on the unit circle S = { x, y R 2 ; x 2 + y 2 }. Moreover, a function f : S R is a trigonometric polynomial of degree n if and only if there exists a polynomial P : R 2 R such that deg P n and f = P S, i.e., Consider the unit sphere fθ = P cos θ, sin θ, θ [, 2π]. S 2 = { x R 3 ; x = }, equipped with the normalized area element 4π da given in spherical coordinates θ, ϕ, θ [, 2π], ϕ [, π] by sin ϕ da = 4π 4π dϕdθ. For each natural number n we denote by U n the subspace of C S 2 consisting of the restrictions to S 2 of the polynomials P x, y, z of degree n. Let us point out that two

16 6 LVU. NCOLAESCU Figure 3. The graph of the restriction to S 2 of a random degree 7 polynomial. different polynomials in the variables x, y, z can have identical restrictions to S 2. For example the polynomial F x = x 2 = x 2 + y 2 + z 2 has the same restriction to S 2 as the constant polynomial. n any case, one can show see e.g. [5] that dim U n = n 2. The zero set of a generic smooth function f : S 2 R is -dimensional -submanifold of S 2 and thus it is a disjoint union of circles. We can think of a function f : S 2 R as an altitude function describing the altitude fp of a point p S 2 relative to the sea-level and we can visualize as defining relief on a spherical shaped planet; see Figure 3. From this point of view, the zero set describes the shore lines of this relief; see Figure 4. The energy landscape of a smooth function f : S 2 R concerns the location of the critical points and its critical values. Recall that the critical points of f are the points where θ f = ϕ f =. f p is a critical point of f, then the value fp of f at p is called a critical value of f. The coordinates of the critical points indicate the locations of the mountain tops, the mountain passes and of the sea bottoms. The critical values record the altitudes at these special points. The space U n is equipped with an inner product u, v = 4π S 2 uvda, u, v U n. This defines a Gaussian probability measure Γ n on U n given by Γ n du = 2π n2 /2 e u 2 2 λdu, where λ denotes the canonical Lebesgue measure on U n defined by the above inner product. f we fix an orthonormal basis e i of U n, then we can represent a random polynomial u in

17 ON THE KAC-RCE FORMULA 7 Figure 4. The zero set of the restriction to S 2 of a random degree 7 polynomial. the ensemble U n, Γ n as a linear combination u = i X i e i where X i are independent standard normal random variables. For each open set O S 2, any open interval R and any smooth function f : S 2 R we denote by Cf, O the number of critical points of f in O, and by Df, the number of critical values of f in. We obtain in this fashion random variables U n u Cf, O, Df, Z { }. We denote by C n O and respectively D n their expectations. Using a higher dimensional version of the Kac-Rice formula [, 3] we proved in [8] that C n O 2n2 3 area O as n The behavior of D n is a bit more complicated, but in [9], we described is large n behavior by relating it to statistical invariants of certain ensembles of random, symmetric 3 3 matrices. Appendix A. A probabilistic dictionary For v > and m R we denote by γ v,m the probability measure on R described by γ v,m dx = e x m2 2v dx. 2πv

18 8 LVU. NCOLAESCU We will refer to it as the Gaussian measure on R with mean m and variance v. As v the measure γ v,m converges weakly to the Dirac measure concentrated at m for this reason we set γ,m := δ m. A probability measure µ on R is called Gaussian if v, m R : µ = γ v,m. f above m =, then we say that µ is a centered gaussian measure. Suppose that Ω, A, P is a probability space and X : Ω R is a random variable, i.e. a measurable function. The random variable X is called Gaussian or normal if the push forward X # P is a Gaussian measure γ v,m on R. We write this as X Nv, m. When m = we say that X is a centered Gaussian variable and we write this X Nv. f X Nv, m, then m is the mean or expectation of X m = EX := XωP dω, and v is the variance of X, v = varx = Ω Ω Xω m 2P dω. Let us observe that if X, X 2 : Ω, A, P R are two independent normal variables, X i Nv i, m i, i =, 2, and c, c 2 R, then c X + c 2 X 2 N c 2 v + c 2 2v 2, c m + c 2 m 2. Suppose that U is a finite dimensional real vector space. Denote by B U the σ-algebra of Borel subsets of U. A centered Gaussian measure on U is a probability measure Γ on B U such that any linear function ξ : U R is a centered Gaussian random variable with variance vξ. Such a measure defines a symmetric bilinear form on the dual U C = C Γ : U U R, Cξ, ξ 2 = E Γ ξ ξ 2 = ξ uξ 2 uγdu. Clearly C Γ is nonnegative definite. The bilinear form C Γ is called the covariance form of the Gaussian measure Γ. The centered Gaussian measure Γ is completely determined by its covariance form. Fix an inner product, on U with norm. Then we can identify U with U and a symmetric, nonnegative definite bilinear form C : U U R with a symmetric, nonnegative definite operator C : U U. The inner product determines a canonical centered Gaussian measure Γ on U defined by Γ du = 2π 2 dim U e u 2 2 du. ts covariance form is given by the identity operator U U. For an arbitrary symmetric nonnegative definite operator C : U U we set Γ C = C 2 # Γ = the pushforward of Γ via the linear map C 2 : U U. Then Γ C is the centered Gaussian measure on U with covariance form C. f C is invertible, then Γ C du = e 2 C u,u du. det 2πC U

19 ON THE KAC-RCE FORMULA 9 References [] R. Adler, R.J.E. Taylor: Random Fields and Geometry, Springer Monographs in Mathematics, Springer Verlag, 27. [2] L. Arnold: Über die Nullstellenverteilung zufälliger Polynome, Math. Z [3] J.-M. Azaïs, M. Wschebor: Level Sets and Extrema of Random Processes, John Wiley & Sons, 29. [4] A.T. Barucha-Reid, M. Sambandhan: Random Polynomials, Academic Press, 986. [5] M. Das: Real zeros of a random sum of orthogonal polynomials, Proc. A.M.S., 2797, [6] M. Das: Real zeros of a class of random algebraic polynomials, J. ndian Math. Soc , [7] A. Edelman, E. Kostlan: How many zeros of a random polynomial are real?, Bull. A.M.S. New Series, 32995, No., -37. [8] P. Erdös, A.C. Offord: On the number of real roots of a random algebraic equation, Proc. London Math. Soc. 6956, [9]. Fáry: Sur la courbure totale d une courbe gauche faisant un nord, Bull. Soc. Math. France, 77949, [] P.J. Forrester, G. Honner: Exact statistical properties of the zeros of complex random polynomials, J.Phys. A: Math. Gen , [].A. bragimov, N.B Maslova: On the expected number of real zeros of random polynomials. Coefficients with zero means, Theory Prob. Appl., 697, [2] M. Kac: On the average number of real roots of a random algebraic equation, Bull. A.M.S., 49943, [3] N. B. Maslova: On the variance of the number of real roots of random polynomials, Theory Probab. Appl., 9974, [4] J.W. Milnor: On the total curvature of knots, Ann. Math., 5295, [5] C. Müller: Analysis of Spherical Symmetries in Euclidean Spaces, Appl. Math. Sci. vol. 29, Springer Verlag, 998. [6] H. Nguyen, O. Nguyen, V. Vu: On the number of real roots of random polynomials, arxiv: [7] L.. Nicolaescu: Statistics of linear families of smooth functions on knots, arxiv [8] : Critical sets of random smooth functions functions on compact manifolds, arxiv:.599 [9] Complexity of random smooth functions on compact manifolds, ndiana Univ. Math. J., 6324, [2] D.. Šparo, M.G.Šur: On the distributions of roots of random polynomials, Vestn. Mosk. Univ. Ser. Mat. Mekh, 962, no.3, [2] M. Sodin, B. Tsirelson: Random complex zeroes,. Asymptotic normality, srael J. Math., 4424, [22] G. Szegö: Orthogonal polynomials, Amer. Math. Soc., Colloquium Publications, vol.23, Reprint. Department of Mathematics, University of Notre Dame, Notre Dame, N address: nicolaescu.@nd.edu URL: