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.

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