Last October, Romain Dujardin gave a nice talk at Bourbaki seminar about the equidistribution of Fekete points, pluripotential theory and the works of Robert Berman, Sébastien Boucksoum and David Witt Nyström(including this article here). The video of Dujardin’s talk (in French) is available here and the corresponding lecture notes (also in French) are available here.

In the sequel, I will transcript my notes for Dujardin’s talk (while referring to his text for all omitted details). In particular, we will follow his path, that is, we will describe how a question related to polynomial interpolation was solved by complex geometry methods, but we will not discuss the relationship of the material below with point processes.

Remark 1As usual, any errors/mistakes in this post are my sole responsibility.

**1. Polynomial interpolation and logarithmic potential theory in one complex variable**

**1.1. Polynomial interpolation**

Let be the vector space of polynomials of degree in one complex variable. By definition, .

The classical polynomial interpolation problem can be stated as follows: given points on , can we find a polynomial with prescribed values at ‘s? In other terms, can we invert the evaluation map , ?

The solution to this old question is well-known: in particular, the problem can be explicitly solved (whenever the points are distinct).

What about the effectiveness and/or numerical stability of these solutions? It is also well-known that they might be “unstable” in many aspects: for instance, the inverse of starts to behave badly when some of the points get close together, a small error on the values might lead to huge errors in the polynomial , Runge’s phenomenon shows that certain interpolations about equidistant point in are highly oscillating, etc.

This motivates the following question: are there “optimal” choices for the points (leading to “minimal instabilities” in the solution of the interpolation problem)?

This vague question can be formalized in several ways. For instance, the interpolation problem turns out to be a linear algebra question asking to *invert* an appropriate Vandermonde matrix and, *a fortiori*, the calculations will eventually oblige us to *divide* by an adequate determinant . Hence, if we denote by , , the base of monomials of , then we can say that an optimal configuration *maximizes* the modulus of Vandermonde’s determinant

Of course, this optimization problem has a trivial solution if we do not impose constraints on . For this reason, we shall fix some compact subset and we will assume that .

Definition 1A Fekete configuration is a maximum of

Definition 2The -diameter of is

where is a Fekete configuration.

It is not hard to see that , i.e., is a decreasing sequence.

Definition 3is the transfinite diameter.

The transfinite diameter is related to the *logarithmic potential* of .

**1.2. Logarithmic potential**

In the one-dimensional setting, the equidistribution of Fekete configurations towards an equilibrium measure was established by Fekete and Szegö.

Theorem 4 (Fekete, Szegö)If and is a sequence of Fekete configurations, then the sequence of probability measures

converges in the weak-* topology to the so-called equilibrium measure of .

*Proof:* Let us introduce the following “continuous” version of Fekete configurations. Given a measure on , its “energy” is

so that if we *forget* about the “diagonal terms” , then . Recall that the capacity of is where (and stands for the space of probability measures on ).

Theorem 5 (Frostman)Either is polar, i.e., for all or there is an unique with .

We are not going to prove this result here. Nevertheless, let us mention that an important ingredient in the proof of Frostman’s theorem is the logarithmic potential associated to : it is a subharmonic function whose (distributional) Laplacian is . A key feature of the logarithmic potential is the fact that if , then for -almost every : observe that this allows to conclude the uniqueness of because it would follow from that is harmonic, “basically” zero on , and near infinity.

Anyhow, it is not hard to deduce the equidistribution of Fekete configurations towards from Frostman’s theorem. Indeed, let be and consider the modified energy . A straighforward calculation (cf. the proof of Théorème 1.1 in Dujardin’s text) reveals that if is a converging subsequence, say , then .

**1.3. Two remarks**

The capacity of admits several equivalent definitions: for instance, the quantities

form a submultiplicative sequence (i.e., ) and the so-called Chebyshev constant

coincides with . In other terms, the capacity of is the limit of certain geometrical quantities associated to a natural norm on the spaces of polynomials .

Also, it is interesting to consider the maximization problem for *weighted* versions

of the energy of measures.

As it turns out, these ideas play a role in higher dimensional context discussed below.

**2. Pluripotential theory on **

Denote by the space of polynomials of degree on complex variables: it is a vector space of dimension as .

Let be a compact subset of and consider . Similarly to the case , the interpolation problem of inverting the evaluation map involves the computation of the determinant where is the base of monomials. Once again, we say that a collection of points in maximizing the quantity is a Fekete configuration and the transfinite diameter of is where

is the –*diameter* of .

Given the discussion of the previous section, it is natural to ask the following questions: do Fekete configurations equidistribute? what about the relation of the transfinite diameter and pluripotential theory?

A first difficulty in solving these questions comes from the fact that it is not easy to produce a “continuous” version of Fekete configurations via a natural concept of energy of measures having all properties of the quantity in the case .

A second difficulty towards the questions above is the following: besides the issues coming from pairs of points which are too close together, our new interpolation problem has new sources of instability such as the case of a configuration of points lying in an algebraic curve. In particular, this hints that some techniques coming from complex geometry will help us here.

The next result provides an answer (comparable to Frostman’s theorem above) to the second question:

Theorem 6 (Zaharjuta (1975))The limit of exists. Moreover, if is not pluripolar, then .

Here, we recall that a pluripolar set is defined in the context of pluripotential theory as follows. First, a function on a open subset is called plurisubharmonic (psh) whenever is upper semicontinuous (usc) and is subharmonic for any holomorphic curve. Equivalently, if , then is psh when is usc and the matrix of distributions is positive-definite Hermitian, i.e., . Next, is *pluripolar* whenever where is a psh function.

An important fact in pluripotential theory is that the positive currents can be multiplied: if are bounded psh functions, then the exterior product can be defined as a current. In particular, we can define the *Monge-Ampère operator* on the space of bounded psh functions.

Note that is a positive current of maximal degree, i.e., a positive measure. This allows us to define a candidate for the equilibrium measure in higher dimensions in the following way.

Let

be the so-called *Lelong class* of psh functions. Given a compact subset , let

Observe that is a natural object: for instance, it differs only by an additive constant from the logarithmic potential of the equilibrium measure of when . Indeed, this follows from the key property of the logarithmic potential (“ implies for -almost every ”) mentioned earlier and the fact that is a subharmonic function which essentially vanishes on .

In general, is *not* psh. So, let us consider the psh function given by its *usc regularization* . Note that , so that we can define the *equilibrium measure* of as

It is worth to point out that can be recovered from the study of polynomials (in a similar way to our discussion of the Chebyshev constant in the previous section). More concretely, we have for all and a result of Siciak ensures that

Finally, note that this discussion admits a weighted version where the usual Euclidean norm on is replaced by , the determinant is replaced by , etc.

**3. Bernstein–Markov property**

Let be a probability measure on giving zero mass to all pluripolar subsets. Since algebraic subvarieties are pluripolar (because they are included in the locus of some polynomial ), we have that gives no weight to algebraic subvarieties. Hence, a polynomial with must vanish, i.e., .

It follows that is *equivalent* to the uniform norm on the finite-dimensional vector spaces , i.e., there is a (best) constant such that

for all .

Definition 7We say that has the Bernstein–Markov property if .

The following result ensures that one can “always” work with measures enjoying the Bernstein–Markov property:

Theorem 8 (Bloom–Levenberg)Every non-pluripolar compact subset supports a probability measure with the Bernstein–Markov property.

Moreover, if where is an open subset of with smooth boundary, then the equilibrium measure has the Bernstein–Markov property (see this paper here):

Theorem 9 (Siciak)If is regular (i.e., is continuous), then has the Bernstein–Markov property.

The Bernstein–Markov property is useful because it allows to interpret the transfinite diameter as the growth rate of certain volumes in the spaces of polynomials.

More concretely, let be a non-pluripolar set and a probability measure with the Bernstein–Markov property.

Lemma 10One has

with .

Lemma 11One has

Since the volumes of the unit balls of a vector space equipped with two Hermitian structures and are given by a Gram determinant, i.e.,

where is an orthonormal basis with respect to , one gets from this fact and the previous two lemmas (whose [elementary] proofs are omitted) that

where is the Haar measure on the torus and is the unit ball of with respect to the Hermitian structure . (Here, it is natural to introduce because the base of monomials of is *orthonormal* with respect to .)

In summary, we saw that the transfinite volume is related to the quotients of volumes of unit balls in the spaces of polynomials. In the last part of this post, we will investigate the problem of understanding quotients of volumes of unit balls in the more general setting of sections of positive vector bundles over complex manifolds.

**4. Global pluripotential theory**

Let be a compact complex manifold of dimension . Suppose that is an ample line bundle, i.e., there is a smooth metric which is strictly psh (in the sense that any local trivialization of has the property that with ).

By the asymptotic Riemann-Roch theorem, the dimensions of the spaces of sections of the th tensor power of verify

as , where is the volume of .

Example 1For the line bundle over the complex projective space , the space identifies with (in a given chart of )

Let us fix a reference metric whose curvature form is denoted by . Note that all metrics on have the form where is a function. We say is semi-positive (i.e., ) whenever is psh in the sense that .

Given a smooth positive measure on and a metric on , one has a natural -structure on , namely

where stands for the metric induced on by .

Theorem 12 (Berman–Boucksoum)Let , be two smooth positive measures and , two metrics on . Then,

where is the Monge-Ampère energy of .

(Here, we insist on the quotients of volumes because the volumes are well-defined only up to a multiplicative constant.)

The Monge-Ampère energy is an interesting functional playing the role of a primitive of the Monge-Ampère operator in the sense that

The idea of proof of Berman–Boucksoum theorem above goes along the following lines (cf. this article here for more details). We write where is a path connecting and . The quotient of volumes in the left-hand side is a Gramm determinant (similar to (1)) with respect to an appropriate choice of basis of sections of and its derivative is the so-called Bergman kernel. In particular, the desired result comes from a comparison between Monge-Ampère operators and Bergman kernels.

For some applications of this result (e.g., to the study of Fekete configurations), it is important to improve this theorem to the case of *general* measures and . As it turns out, this is done by considering the quotients of volumes with respect to norms on induced by the metrics and the Monge-Ampère energies of certain functions playing the analog role of the functions introduced above in the context of compact subset .

Anyhow, let us close this post by explaining how this technology can be used to get the equidistribution of Fekete points in higher dimensional settings.

**5. Equidistribution of Fekete configurations**

We saw that the transfinite diameter is related to the quotient of two volumes (cf. (2)). In order to exploit this fact, we need the following result of Berman–Boucksoum.

Theorem 13 (Berman–Boucksoum)The Monge-Ampère energy is differentiable with respect to , namely

Remark 2The Monge-Ampère measure is the candidate to the equilibrium measure.

Besides the previous theorem, we also need a technical fact about concave functions of the real line:

Lemma 14Let be a sequence of (differentiable) concave functions defined in a neighborhood of . If is a (differentiable) concave function on such that

and

then .

At this point, the idea is to derive the equidistribution of Fekete configurations towards an equilibrium measure in higher dimensional settings from the previous lemma applied to the geometric functionals

and

because a straightforward calculation reveals that

and Theorem 13 and (the generalization to arbitrary measures of) Theorem 12 yield

Remark 3After the end of the talk, I asked Romain about speed of equidistribution of Fekete points and he pointed out to me that such results were recently obtained by T.-C. Dinh, X. Ma and V.-A. Nguyên.

## Leave a Reply