Expected energy of zeros of elliptic polynomials

In 2011, Armentano, Beltr\'an and Shub obtained in \cite{ABS11} a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different approach which allows us to extend the study to the Riesz energies and to compute the expected separation distance.


Introduction and Main results
Elliptic polynomials, also called Kostlan-Shub-Smale or SU (2) polynomials, are defined by where a n are i.i.d.random variables with standard complex Gaussian distribution.These polynomials appeared first in the mathematical physics literature [BBL92, BBL96,Han96] and were readily studied from a mathematical point of view [Kos93,SS93].One reason for the interest in these polynomials is that the random point process on S 2 given by the stereographic projection of the roots of elliptic polynomials is invariant through rotations.Moreover, it is the unique point process given by zeros of random analytic functions with this property [Sod00].Among its many interesting properties, especially relevant are the connections, studied in [SS93], with well conditioned polynomials and with minimal logarithmic energy points.
The Riesz or logarithmic energy of a set of N different points x 1 , . . ., x N on the unit sphere where f s (r) = r −s for s = 0 and f 0 (r) = − log r are, respectively, the Riesz and logarithmic potentials.We denote the extremal (minimal or maximal) energy attained by a set of N points on the sphere by E s (N ) = min x1,...,x N ∈S 2 E s (x 1 , . . ., x N ) if s ≥ 0, max x1,...,x N ∈S 2 E s (x 1 , . . ., x N ) if s < 0.
The condition number of a univariate polynomial, defined by Shub and Smale, is a measure of how much the roots of a polynomial change when perturbing the coefficients.It was shown in [SS93b] that points of almost minimal logarithmic energy, s = 0, are the roots of well conditioned polynomials.In [SS93], the authors also proved that, with high probability, elliptic polynomials are well conditioned, see [BEMOC21,BL20] for a deterministic example.It was therefore natural to study the expected energy of the zeros of elliptic polynomials.This was done in [ABS11], where the authors obtained the following closed expression for the expected logarithmic energy of random points x 1 , . . ., x N ∈ S 2 , images by the stereographic projection of zeros of elliptic polynomials, (1) The asymptotic expression above is indeed very close to the minimal logarithmic energy of N points on the sphere, see Section 4. Working in a more general setting, in [Zho08,ZZ10] the same expression (1) was obtained but with a o(N ) remainder.Our main result is an extension of the above result (1) to the Riesz s-energies for s < 4.
By considering two terms of the asymptotic expansion of the Hurwitz Zeta function for |a| < 1 and s = 1 [DLMF, 25.11.10] and taking m = 1 in (2) we get, for 0, 2 = s < 4, when N → ∞, where Remark 1.The result above for the expected Riesz energy allows us to compare the zeros of elliptic polynomials with other point processes, for example in terms of expected p-moments of averages.Indeed, from Khintchine's inequality [KK01, Theorem 3], it follows that when x 1 , . . ., x N are uniform i.i.d. points on the sphere S 2 and 1 ≤ p < ∞.For points drawn from the spherical ensemble, for which there is repulsion between points, it follows from (8 and the results about the expected Riesz energy s = −2 in [AZ15] that the expected 2-moment is bounded.Hence, for the spherical ensemble E N i=1 x i p is bounded for 1 ≤ p ≤ 2, and numerical simulations suggest that the same holds for p > 2. In our case, for zeros of elliptic polynomials mapped to the sphere by the stereographic projection, it follows from (5) that for N → +∞, and the average p-moments for 1 ≤ p ≤ 2 converge to zero.Again, numerical simulations suggest the same behavior for p > 2. It is well known that minimal logarithmic points have center of mass in the center of the sphere, i.e. have zero dipole, [BHS19, Corollary 6.7.5], [BBP94].Therefore, the behavior of the expected p-moments matches the particularly low logarithmic energy of zeros of elliptic polynomials.For the comparison with minimal and expected energies of other point processes, see discussion in Section 4. In our last result, we compute a closed expression for the expected separation distance between points drawn from zeros of elliptic polynomials.The separation distance of and its counting version by G(t, X N ) = {i < j : Recall that energy minimizers have a separation distance of order N −1/2 , [BHS19, Section 6.9].
Theorem 1.2.Let X N be a set of N −points drawn from zeros of elliptic polynomials mapped to the sphere by the stereographic projection.Then , and moreover and therefore, as in the harmonic case, see [BMOC16], an N -tuple drawn from the zeros of elliptic polynomials likely satisfies sep(X N ) = Ω(N −3/4 ), Figure 2. See also [AZ15, Corollary 1.6] for the analogue result for the spherical ensemble.1.1.Organization of the paper.In section 2 we compute the 2-point intensity function of our point process and explain how to compute the expected energy.Section 3 contains the proof of our main result, Theorem 1.1.In section 4 we deduce some bounds for the extremal energy and compare our bounds with previous results.Finally, in section 5 we prove Theorem 1.2 about separation.

Intensity function and expected Riesz energy
In this section we compute the 2-point intensity function of the random point process on S 2 corresponding to the stereographic projection of the roots of random elliptic polynomials where a n are i.i.d.random variables with standard complex Gaussian distribution.Let F (x, y) be a measurable function defined on S 2 × S 2 whose variables will be considered in C through the stereographic projection, i.e.F (z, w) = F (x(z), y(w)), with the points x, y ∈ S 2 corresponding to z, w ∈ C. By Campbell's formula, if x 1 , . . .x N ∈ S 2 are the images of the zeros z 1 , . . ., z N of elliptic polynomials, then with ρ 2 (z, w) the 2-point intensity function given by [HKPV09, Corollary 3.4.2] where A, B, C are the 2 × 2 matrices It is easy to see that when F is rotational invariant we get Therefore, it is enough to compute ρ 2 (z 1 , z 2 ) for z 1 = z ∈ C and z 2 = 0.The matrices in (14) are then and we obtain see [Han96] and Figure 3 where one can notice that this point process is not determinantal ([HKPV09, p.83]).
Using the relation with the chordal metric we get and for s = 0 (16) In the logarithmic case, one can compute directly a primitive function that leads to the correct energy (1).However, we will compute the expected logarithmic energy as the limit of the Riesz case at s = 0.

Proof of theorem 1.1
In this section we prove first our general result (2) with the auxiliary Proposition 3.1.Then we prove the cases (4),(5) and finally (3).
Proof.To simplify the notation we write The integrand is equivalent to x −2 at infinity, which is integrable, and to x 1−s/2 at x = 0, which is integrable iff 1 − s/2 > −1.Then, the energy will be finite iff s < 4. Now let us compute the integral.We take r = s/2 for simplicity, so we will be assuming r < 2 throughout the proof.Using that k+1) for x > 1 and the fact that all the terms are positive, we get Using the following integral representation for the beta function (see [GR07,8.380 (3)]), To compute A k we integrate by parts. Then or, in terms of the gamma function, provided that r = 1.The case r = 1 will be studied as the limit r → 1.
Therefore, for r = 1, writing all together The sums get simplified by using the property Γ(z + 1) = zΓ(z) and changing the indices in such a way that all quotients have the form Γ(N k + 1)/Γ(N k + 2 − r) Taking the asymptotic expansion of the terms in M as M → ∞, we get Applying Proposition 3.1 below we obtain the following expression for every r = 0, 1 with r < 2 Writing the expression in terms of s = 2r yields the result (2).Now we prove (4), i.e. r = 1, from the case r = 1.By continuity, the evaluation of the integral at the beginning of (18) can be performed by taking the limit r → 1 in A k , B k , C k , that is, in both sums in (22).The only tricky limit is the first one.It can be computed using the asymptotic expansion for γ → 0, where a will be a natural number.Considering γ = 1 − r, and we get from ( 22) The first sum in (24) can be rewritten as while the second becomes We will use the functional relation ψ(x + 1) = ψ(x) + 1 x for the digamma function, which allows us to obtain, for instance, Using this we get We can simplify Σ 1 with the same property.Since Therefore, From the relation [GR07, 8.365 (6)], Summation by parts gives for every 1 ≤ j ≤ N and then Using the asymptotic expansion Then Finally, using [Bla15, (B.11)], we get (4) To compute E[E −2n ] and E[E 0 ], we start observing that for r < 1 formula (23) yields since both sums are convergent in this case.Using the expression of the beta function in terms of gamma function and the monotone convergence theorem, we get For r = −n, the energy is To compute I 1 and I 2 we will use the following integral representation [GR07, 8.361 (7)] for the digamma function for any a > −1.Then where we have used n m=0 n m (−1) m = 0 in the second and last equality.Applying ψ(x + 1) = ψ(x) + 1/x, and it is trivial to check that the second sum equals −1/(n + 1).
The integral I 2 can be computed in a similar way where the second sum is In order to compute E[E 0 ], i.e. formula (3) from [ABS11], we take the derivative of E[E s ] at s = 0. Consider the continuous function for r = 0, N 2 − N, for r = 0, where r = 0 matches the Riesz 0-energy, which trivially is N 2 −N for any configuration of points.Then Since g (0) exists, we can derive it by restricting to r < 0 where according to (25). Then By continuity, we also have lim r→0 − g(r) = g(0) = N 2 − N , so we deduce that (28) Therefore, where we have applied (27).
It remains to compute the limit where the limit of the last integral is justified by monotone convergence theorem.Using (27), we obtain where we have used that From (29) we finally get Proof.We will use the following Fields' approximation for the quotient of gamma functions, see [DLMF, Eq. 5.11.14] or [Fie66] Γ(z + a) The sums F and I are convergent and G can be written as (see [DLMF, Eq. 25.11.5]) The same formula holds to approximate D for r < 1 (33) where the last integral converges for 1 < r < 2 when M → +∞ to ζ 1 − r, 2−r 2N , see [DLMF, Eq. 25.11.26].
Applying this limit on (35) we get the desired result.

Bounds for the minimal energy asymptotic expansion
We will start this section recalling some known results, and some conjectures, about the asymptotic expansion of the extremal energy E s (N ) attained by a set of N points on the sphere S 2 .For a more complete picture see [BHS19].
The current knowledge about the asymptotic expansion of the minimal energy is far from complete even in S 2 , but for s ≤ −2 the situation is well known.Indeed, the minimizers of the Riesz energy for s < −2 are points placed at each of the two endpoints of some diameter (for even N ), [Bjo56], and for s = −2, formula (8) shows that any configuration with center of mass at the origin attains the maximum 2N 2 .
For 0 < |s| < 2, it is known that there exist c, C > 0 (depending on s) such that (36) Wag90,Wag92] and [Bra06,AZ15] for improvements in the value of the constants leading to the bounds (37) which were obtained with the bound given by the expected energy of random points from the spherical ensemble [AZ15].
In the boundary case s = 2, it was shown in [BHS12, Proposition 3] that and the upper bound was improved in [AZ15] to where γ is the Euler-Mascheroni constant.
For the logarithmic potential, it is known that there exists a constant, C log , such that for which see [BS18,Lau21] and [BL21] for a recent direct computation of the lower bound.The upper bound for C log has been conjectured to be an equality by two different approaches [BHS12,BS18].
For −2 < s < 4, s = 0, the asymptotic expansion of the optimal Riesz s-energy has been conjectured in [BHS12] to be, for s = 2, (40) where ζ Λ2 (s) is the zeta function of the hexagonal lattice, while for s = 2 the conjectured expansion is It is clear that the minimal energy is always bounded by the expected energy with respect to a given random configuration.Therefore, one can bound the asymptotic expansion of the minimal energy by the asymptotic expansion of the expected energy.This idea was used in [ABS11] to get bounds for the minimal logarithmic energy using (1) and in [AZ15] to get (37) and (38).For other computations of expected energies in different settings, see [BS13, BMOC16, BE18, MOC18, BE19, BF20, BDFS22, ADGMS22].From our main result, Theorem 1.1, we obtain the asymptotic expansion (6) which is close to the conjectured expansion for the minimal energy, see figure 4, and we can prove the following bounds.
Since the constant C(s) is negative, we can choose δ = − C(s) to obtain the result.
For s = 2, the energy is (4): We can rewrite the sum as (43) in such a way that the term corresponding to j = N in the first sum is well-defined.Let us apply the Euler-Maclaurin formula to f (x) = g(x/N ), with g where B j are the Bernoulli numbers and R A N is the remainder term, that satisfies We get where H N is the N -th harmonic number.Its expansion as N → ∞, see [Boa77], is With these expansions, formula (43) reads Plugging this into the formula (4), we obtain (46) . The blue curve is given by the conjectured valued for the second order constant, (40).The green and yellow curves corresponds to [RSZ94] and [AZ15], respectively, and the red is our constant (7).

Proof of theorem 1.2
Proof.Since the function F (p, q) = 1 {|p−q|≤t} is rotational invariant, we can apply the formula (15) where we have applied the change of variables r = √ x.As in the proof of Theorem 1.1, we use the identity 1 (x−1) 3 = 1 2 ∞ k=2 k(k − 1)x −(k+1) for x > 1 to get The expression is the same than (18) with r = 0, but changing the upper limit of integration.We can take advantage of our previous computations using the following representation for the incomplete beta function Remember that our goal is to check that g N (s) ≥ 0 for s ≥ 0. In fact, we will see that the coefficients of this polynomial are all positive for any N ≥ 2. To prove this, let us successively apply the identity n k = n−k+1

Figure 1 .
Figure 1.Plot of 4ζ(3)/N and realizations of | N i=1 x i | 2 for natural N up to 1000.

Figure 2 .
Figure 2. x marks correspond to the values of the minimal separation for realizations of N elliptic zeros for natural N from 10 up to 1000.The continuous graph are cN −3/4 for c = 1.89 (yellow) and 3.27 (brown): using Chebyshev's inequality at least 90% of the realizations are above yellow and at least 10% above brown.