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APPEAREDINBULLETINOFTHE AMERICANMATHEMATICALSOCIETY Volume32,Number1,January1995,Pages1–37 HOW MANY ZEROS OF A RANDOM POLYNOMIAL ARE REAL? 5 9 9 ALAN EDELMAN AND ERIC KOSTLAN 1 n a J Abstract. We provide an elementary geometric derivation of the Kac inte- 1 gral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that ] the expected number of real zeros is simply the length of the moment curve A (1,t,...,tn)projectedontothesurfaceoftheunitsphere,dividedbyπ. The C probability density of the real zeros is proportional to how fast this curve is . tracedout. h WethenrelaxKac’sassumptionsbyconsideringavarietyofrandomsums, at series,anddistributions,andwealsoillustratesuchideasasintegralgeometry m andtheFubini-Studymetric. [ Contents 1 v 1. Introduction 4 2. Random polynomials and elementary geometry 2 2.1. How fast do equators sweep out area? 2 2.2. The expected number of real zeros of a random polynomial 1 0 2.3. Calculating the length of γ 5 2.4. The density of zeros 9 2.5. The asymptotics of the Kac formula / h 3. Random functions with central normal coefficients t 3.1. Random polynomials a m 3.2. Random infinite series 4. Theoretical considerations : v 4.1. A curve with more symmetries than meet the eye i 4.2. Geodesics on flat tori X 4.3. The Kac matrix r a 4.4. The Fubini-Study metric 4.5. Integral geometry 4.6. The evaluation mapping 5. Extensions to other distributions 5.1. Arbitrary distributions 5.2. Non-central multivariate normals: Theory Receivedbytheeditors November22,1993,and,inrevisedform,July25,1994. 1991 Mathematics Subject Classification. Primary34F05; Secondary30B20. Keywordsandphrases. Randompolynomials,Buffonneedleproblem,integralgeometry,ran- dompowerseries,randommatrices. SupportedbytheAppliedMathematicalSciencessubprogramoftheOfficeofEnergyResearch, U.S.DepartmentofEnergy,underContractDE-AC03-76SF00098. (cid:13)c1995 American Mathematical Society 1 2 ALAN EDELMAN AND ERIC KOSTLAN 5.3. Non-central multivariate normals: Applications 6. Eigenvalues of random matrices 6.1. How many eigenvalues of a random matrix are real? 6.2. Matrix polynomials 7. Systems of equations 7.1. The Kac formula 7.2. A random polynomial with a simple answer 7.3. Random harmonic polynomials 7.4. Random power series 7.5. Random entire functions 8. Complex zeros 8.1. Growth rates of analytic functions 8.2. A probabilistic Riemann hypothesis 9. The Buffon needle problem revisited Acknowledgments References 1. Introduction Whatis the expectednumber ofrealzerosE ofarandompolynomialofdegree n n? If the coefficients are independent standard normals, we show that as n , →∞ 2 2 E = log(n) + 0.6257358072... + + O(1/n2) . n π nπ The 2 logntermwasderivedbyKacin1943[26],whoproducedanintegralformula π for the expected number of real zeros. Papers on zeros of random polynomials include [3], [16], [23], [34], [41, 42] and [36]. There is also the comprehensive book of Bharucha-Reid and Sambandham [2]. We will derive the Kac formula for the expected number of real zeros with an elementary geometric argument that is related to the Buffon needle problem. We presentthe argumentin a manner such that precalculus levelmathematics is sufficient for understanding (and enjoying) the introductory arguments, while elementary calculus and linear algebra are sufficient prerequisites for much of the paper. Nevertheless, we introduce connections with advanced areas of mathematics. Aseeminglysmallvariationofouropeningproblemconsidersrandomnthdegree polynomials with independent normally distributed coefficients, each with mean zero, but with the variance of the ith coefficient equal to n (see [4], [31], [46]). i This particular random polynomial is probably the more natural definition of a (cid:0) (cid:1) random polynomial. It has E =√n n real zeros on average. Asindicatedinourtableofcontents,theseproblemsserveasthedeparturepoint for generalizations to systems of equations and the real or complex zeros of other collections of random functions. For example, we consider power series, Fourier series, sums of orthogonal polynomials, Dirichlet series, matrix polynomials, and systems of equations. Section2beginswithourelementarygeometricderivation. Section3showshow alargeclassofrandomproblemsmaybecoveredinthisframework. InSection4we HOW MANY ZEROS OF A RANDOM POLYNOMIAL ARE REAL? 3 revealwhatisgoingonmathematically. Section5studiesarbitrarydistributionsbut focusesonthenon-centralnormal. Section6relatesrandompolynomialstorandom matrices, while Section 7 extends our results to systems of equations. Complex roots,which are ignoredin the rest of paper, are addressedin Section 8. We relate random polynomials to the Buffon needle problem in Section 9. 2. Random polynomials and elementary geometry Section 2.1 is restricted to elementary geometry. Polynomials are never men- tioned. The relationship is revealed in Section 2.2. 2.1. How fast do equators sweep out area? We will denote (the surface of) the unit sphere centered at the origin in Rn+1 by Sn. Our figures correspond to the case n=2. Higher dimensions provide no further complications. Definition 2.1. If P Sn is any point, the associated equator P is the set of ⊥ ∈ points of Sn on the plane perpendicular to the line from the origin to P. This generalizes our familiar notion of the earth’s equator, which is equal to (north pole) and also equal to (south pole) . See Figure 1. Notice that P is ⊥ ⊥ ⊥ always a unit sphere (“great hypercircle”) of dimension n 1. − Let γ(t) be a (rectifiable) curve on the sphere Sn. Definition 2.2. Let γ , the equators of a curve, be the set P P γ . ⊥ ⊥ { | ∈ } Assume thatγ hasa finite length γ . Let γ to be the area“sweptout”byγ ⊥ ⊥ | | | | (we will provide a precise definition shortly). We wish to relate γ to γ . ⊥ | | | | If the curve γ is a small section of a great circle, then γ is a lune, the area ⊥ ∪ bounded by two equators as illustrated in Figure 2. If γ is an arc of length θ, then our lune covers θ/π of the area of the sphere. The simplest case is θ =π. We thus obtain the formula valid for arcs of great circles, namely, γ γ ⊥ | | = | |. area of Sn π If γ is not a section of a greatcircle, we may approximate it by a union of small great circular arcs, and the argument is still seen to apply. The alert reader may notice something wrong. What if we continue our γ so that it is more than just half of a great circle, or what if our curve γ spirals many timesaroundapoint? Clearly,wheneverγ is notapiece ofa greatcircle,thelunes will overlap. The correct definition for γ is the area swept out by γ(t) , as t ⊥ ⊥ | | varies, counting multiplicities. We now give the precise definitions. P P P P Figure 1. Points P and associated equators P . ⊥ 4 ALAN EDELMAN AND ERIC KOSTLAN γ γ ⊥ Figure 2. The lune γ when γ is a great circular arc. ⊥ ∪ Definition 2.3. The multiplicity of a point Q γ is the number of equators in ⊥ γ that contain Q, i.e., the cardinality of t R∈Q∪ γ(t) . ⊥ ⊥ { ∈ | ∈ } Definition 2.4. We define γ to be the area of γ counting multiplicity. More ⊥ ⊥ | | ∪ precisely, we define γ to be the integral of the multiplicity over γ . ⊥ ⊥ | | ∪ Lemma 2.1. If γ is a rectifiable curve, then γ γ ⊥ (1) | | = | |. area of Sn π Asanexample,considerapointP onthesurfaceoftheEarth. Ifweassumethat the point P is receiving the direct ray of the sun—for our purposes, we consider the sun to be fixed in space relative to the Earth during the course of a day, with rays arriving in parallel—then P is the great circle that divides day from night. ⊥ This greatcircleis knownto astronomersasthe terminator(Figure 3). During the Earth’sdailyrotation,the pointP runsthroughallthe pointsonacircleγ offixed latitude. Similarly, the Earth’srotationgeneratesthe collectionof terminatorsγ . ⊥ The multiplicity in γ is two on a regionbetween two latitudes. This is a fancy ⊥ mathematical way of saying that unless youare too close to the poles, you witness bothasunriseandasunseteveryday! Thesummersolsticeisaconvenientexample. P is on the Tropic of Cancer and Equation (1) becomes 2 (The surface area of the Earth between the Arctic/Antarctic Circles) × The surface area of the Earth The length of the Tropic of Cancer = π (The radius of the Earth) × or equivalently The surface area of the Earth between the Arctic/Antarctic Circles The surface area of the Earth The length of the Tropic of Cancer = . The length of the Equator 2.2. The expected number of real zeros of a random polynomial. What does the geometric argument in the previous section and formula (1) in particular have to do with the number of real zeros of a random polynomial? Let p(x)=a +a x+ +a xn 0 1 n ··· HOW MANY ZEROS OF A RANDOM POLYNOMIAL ARE REAL? 5 P γ Day Night Terminator Figure 3. On the summer solstice, the direct ray of the sun reaches P on the Tropic of Cancer γ. be a non-zero polynomial. Define the two vectors a 1 0 a t 1     a t2 a= 2 and v(t)= .  ..   ..   .   .       a   tn   n        The curve in Rn+1 traced out by v(t) as t runs over the real line is called the moment curve. The condition that x = t is a zero of the polynomial a +a x+ +a xn is 0 1 n ··· precisely the condition that a is perpendicular to v(t). Another way of saying this is that v(t) is the set of polynomials which have t as a zero. ⊥ Define unit vectors a a/ a , γ(t) v(t)/ v(t) . ≡ k k ≡ k k As before, γ(t) corresponds to the polynomials which have t as a zero. ⊥ When n=2, the curve γ is the intersection of an elliptical (squashed) cone and the unit sphere. In particular, γ is not planar. If we include the point at infinity, γ becomes a simple closed curve when n is even. (In projective space, the curve is closed for all n.) The number of times that a point a on our sphere is covered by an equator is the multiplicity of a in γ . This is exactly the number of real zeros ⊥ of the corresponding polynomial. So far, we have not discussed random polynomials. If the a are independent i standardnormals,thenthevectoraisuniformlydistributedonthesphereSn since the joint density function in sphericalcoordinates is a function of the radius alone. 6 ALAN EDELMAN AND ERIC KOSTLAN Figure 4. When n = 2, γ is the intersection of the sphere and cone. The intersectionis a curvethat includes the North Poleand a point on the Equator. What is E the expected number of real zeros of a random polynomial? A n ≡ random polynomial is identified with a uniformly distributed random point on the sphere,soE istheareaofthespherewithourconventionofcountingmultiplicities. n Equation (1) (read backwards!) states that 1 E = γ . n π| | Our question about the expected number of real zeros of a random polynomial is reduced to finding the length of the curve γ. We compute this length in Section 2.3. When n = 2, γ is the intersection of the sphere and cone (Figure 4). The intersection is a curve that includes the North Pole and a point on the Equator. 2.3. Calculating the length of γ. We invoke calculus to obtain the integral formula for the length of γ and hence the expected number of zeros of a random polynomial. The result was first obtained by Kac in 1943. Theorem 2.1 (Kac formula). Theexpectednumberofrealzerosofadegreenpoly- nomial with independent standard normal coefficients is 1 ∞ 1 (n+1)2t2n E = dt (2) n π Z−∞s(t2−1)2 − (t2n+2−1)2 4 1 1 (n+1)2t2n = dt. π Z0 s(1−t2)2 − (1−t2n+2)2 HOW MANY ZEROS OF A RANDOM POLYNOMIAL ARE REAL? 7 Proof. The standard arclength formula is ∞ γ = γ′(t) dt. | | k k Z−∞ We may proceed in two different ways. Method I (Direct approach). To calculate the integrand,we first consider any differentiable v(t):R Rn+1. It is not hard to show that → ′ v(t) [v(t) v(t)]v′(t) [v(t) v′(t)]v(t) γ′(t)= = · − · , v(t) v(t)! [v(t) v(t)]3/2 · · and therefore, p ′ ′ v(t) v(t) γ′(t) 2 = k k v(t) v(t)! · v(t) v(t)! · · [vp(t) v(t)][v′(t) v′(pt)] [v(t) v′(t)]2 = · · − · . [v(t) v(t)]2 · If v(t) is the moment curve, then we may calculate γ′(t) with the help of the k k following observations and some messy algebra: 1 t2n+2 v(t) v(t)=1+t2+t4+ +t2n = − ; · ··· 1 t2 − v′(t) v(t)=t+2t3+3t5+ +nt2n−1 · ··· 1 d 1 t2n+2 t 1 t2n nt2n+nt2n+2 = − = − − ; 2dt 1 t2 (t2 1)2 (cid:18) − (cid:19) (cid:0) − (cid:1) v′(t) v′(t)=1+4t2+9t4+ +n2t2n−2 · ··· 1 d d 1 t2n+2 t2n+2 t2 1+t2n nt2 n 1 2 = t − = − − − − . 4tdt dt 1 t2 (t2 1)3 (cid:18) − (cid:19) − (cid:0) (cid:1) Thus we arrive at the Kac formula: 1 ∞ (t2n+2 1)2 (n+1)2t2n(t2 1)2 E = − − − dt n π (t2 1)(t2n+2 1) Z−∞ p − − 1 ∞ 1 (n+1)2t2n = dt. π Z−∞s(t2−1)2 − (t2n+2−1)2 Method II (Sneaky version). Byintroducingalogarithmicderivative,wecanavoid the messy algebra in Method I. Let v(t) : R Rn+1 be any differentiable curve. → Then it is easy to check that ∂2 (3) log[v(x) v(y)] = γ′(t) 2. ∂x∂y · k k (cid:12)y=x=t (cid:12) Thus we have an alternative expression(cid:12)for γ′(t) 2. (cid:12) k k When v(t) is the moment curve, 1 (xy)n+1 v(x) v(y)=1+xy+x2y2+ +xnyn = − , · ··· 1 xy − 8 ALAN EDELMAN AND ERIC KOSTLAN the Kac formula is then 1 ∞ ∂2 1 (xy)n+1 E = log − dt. n π ∂x∂y 1 xy Z−∞s − (cid:12)y=x=t (cid:12) Thisversionofthe Kacformulafirstappearedin[31]. In(cid:12) Section4.4,werelatethis (cid:12) sneaky approach to the so-called “Fubini-Study” metric. 2.4. The density of zeros. Up until now, we have focused on the length of γ = γ(t) < t < and concluded that it equals the expected number of zeros { |−∞ ∞} on the real line multiplied by π. What we really did, however, was compute the density of real zeros. Thus 1 1 (n+1)2t2n ρ (t) n ≡ πs(t2 1)2 − (t2n+2 1)2 − − is the expected number of real zeros per unit length at the point t R. This is ∈ a true density: integrating ρ (t) over any interval produces the expected number n of real zeros on that interval. The probability density for a random real zero is ρ (t)/E . It is straightforward [26, 27] to see that as n , the real zeros are n n → ∞ concentrated near the point t= 1. ± The asymptotic behavior of both the density and expected number of real zeros is derived in the subsection below. 2.5. TheasymptoticsoftheKacformula. Ashortargumentcouldhaveshown that E 2 logn [26], but since several researchers, including Christensen, Sam- n ∼ π bandham, Stevens, and Wilkins have sharpened Kac’s original estimate, we show here how successive terms of the asymptotic series may be derived, although we will derive only a few terms of the series explicitly. The constant C and the next 1 term 2 were unknown to previous researchers. See [2, pp. 90–91] for a summary nπ of previous estimates of C . 1 Theorem 2.2. As n , →∞ 2 2 E = log(n)+C + +O(1/n2) , n 1 π nπ where 2 ∞ 1 4e−2x 1 C = log(2) + dx 1 π Z0 (sx2 − (1−e−2x)2 − x+1) ! =0.6257358072.... Proof. We now study the asymptotic behavior of the density of zeros. To do this, we make the change of variables t=1+x/n, so ∞ E =4 ρˆ (x) dx , n n Z0 where 1 n4 (n+1)2(1+x/n)2n ρˆ (x)= n nπsx2(2n+x)2 − [(1+x/n)2n+2 1]2 − is the (transformed) density of zeros. Using x n x2 1+ =ex 1 +O(1/n2) , n − 2n (cid:16) (cid:17) (cid:18) (cid:19) HOW MANY ZEROS OF A RANDOM POLYNOMIAL ARE REAL? 9 we see that for any fixed x, as n , the density of zeros is given by →∞ x(2 x) ′ (4) ρˆ (x)=ρˆ (x)+ − ρˆ (x) + O(1/n2) , n ∞ ∞ 2n (cid:20) (cid:21) where 1 1 4e−2x 1/2 ρˆ (x) . ∞ ≡ 2π x2 − (1 e−2x)2 (cid:20) − (cid:21) This asymptotic series cannot be integrated term by term. We solve this problem by noting that χ[x>1] 1 χ[x>1] 1 (5) = + O(1/n2) , 2πx − 2π(2n+x) 2πx − 4nπ where we have introduced the factor 1 if x>1, χ[x>1] ≡(0 if x 1 ≤ to avoid the pole at x=0. Subtracting (5) from (4), we obtain χ[x>1] 1 ρˆ (x) n − 2πx − 2π(2n+x) (cid:26) (cid:27) ′ χ[x>1] x(2 x) 1 = ρˆ (x) + − ρˆ (x) + + O(1/n2) . ∞ ∞ (cid:26) − 2πx (cid:27) ((cid:20) 2n (cid:21) 4πn) We then integrate term by term from 0 to to get ∞ ∞ 1 ρˆ (x) dx log(2n) n − 2π Z0 ∞ χ[x>1] 1 = ρˆ (x) dx+ + O(1/n2) . ∞ − 2πx 2nπ Z0 (cid:26) (cid:27) The theorem immediately follows from this formula and one final trick: we replace χ[x > 1]/x with 1/(x+1) in the definition of C so we can express it as a single 1 integral of an elementary function. 3. Random functions with central normal coefficients Reviewing the discussioninSection 2,we see that we couldomit some members of our basis set 1,x,x2,... ,xn and ask how many zeros are expected to be real { } of an nth degree polynomial with, say, its cubic term deleted. The proof would hardly change. Or we can change the function space entirely and ask how many zeros of the random function a +a sin(x)+a e|x| 0 1 2 are expected to be real—the answer is 0.63662. The only assumption is that the coefficients are independent standard normals. If f ,f ,... ,f is any collection of 0 1 n rectifiable functions, we may define the analogue of the moment curve f (t) 0 f (t) 1 (6) v(t)= . . . .    f (t)   n    10 ALAN EDELMAN AND ERIC KOSTLAN Thefunction 1 γ′(t) isthedensityofarealzero;itsintegraloverRistheexpected πk k number of real zeros. We may relax the assumption that the coefficient vector a=(a ,... ,a )T con- 0 n tains independent standard normals by allowing for any multivariate distribution with zero mean. If the a are normally distributed, E(a) = 0 and E(aaT) = C, i then a is a (central) multivariate normal distribution with covariance matrix C. It is easy to see that a has this distribution if and only if C−1/2a is a vector of standard normals. Since a v(t)=C−1/2a C1/2v(t), · · the density of real zeros with coefficients from an arbitrary central multivariate normal distribution is 1 (7) w′(t) , where w(t)=C1/2v(t), and w(t)=w(t)/ w(t) . πk k k k The expected number of real zeros is the integral of 1 w′(t) . πk k We now state our general result. Theorem 3.1. Let v(t) = (f (t),... ,f (t))T be any collection of differ- 0 n entiable functions and a ,... ,a be the elements of a multivariate normal dis- 0 n tribution withmean zeroandcovariance matrix C. The expectednumberofreal zeros on an interval (or measurable set) I of the equation a f (t)+a f (t)+ +a f (t)=0 0 0 1 1 n n ··· is 1 w′(t) dt, πk k ZI where w is defined by Equations (7). In logarithmic derivative notation this is 1 ∂2 1/2 logv(x)TCv(y) dt. π ∂x∂y y=x=t ZI(cid:18) (cid:19) (cid:0) (cid:1)(cid:12) (cid:12) Geometrically, changing the covariance is the same as changing the inner product on the space of functions. We now enumerate several examples of Theorem 3.1. We consider examples for which v(x)TCv(y) is a nice enough function of x and y that the density of zeros can be easily described. For a survey of the literature, see [2], which also includes the results of numerical experiments. In our discussion of random series, proofs of convergence are omitted. Interested readers may refer to [45]. We also suggest the classicbook ofJ.-P.Kahane[28],where other problemsaboutrandomseries of functions are considered. 3.1. Random polynomials. 3.1.1. The Kac formula. Ifthe coefficients ofrandompolynomialsareindependent standard normal random variables, we saw in the previous section that from 1 (xy)n+1 (8) v(x)TCv(y)= − , 1 xy − we can derive the Kac formula.