A Robin formula for the Fekete-Leja transfinite diameter

We generalize the classical Robin formula to higher dimensions. Mathematics Subject Classification (2000) Primary 32U35 · 32U20 · 14G40; Secondary 31B15 · 31C10 The purpose of this note is to give a formula for the Fekete-Leja transfinite diameter on CN , generalizing the classical Robin formula d∞(E) = e−V(E) for the usual transfinite diameter. We will disengage this from a formula for the sectional capacity proved in arithmetic intersection theory ([5], Theorem 1.1, p. 233). First recall the definition of the Fekete-Leja transfinite diameter for a compact set E ⊂ CN (see [1,16]). Consider the set of monomials zk = z1 1 · · · zN N in the polynomial ring C[z] = C[z1, . . . , zN]. Let (n) ⊂ C[z] be the space of polynomials of total degree at most n, and let K(n) = {k ∈ ZN : ki ≥ 0,k1 + · · · + kN ≤ n} be the index set for the monomial basis of (n). Put qn = #(K(n)) = (n+N n ) . Fixing n, take qn independent vector variables zi = (zi1, . . . , ziN) ∈ CN , i = 1, . . . ,qn. Let k1, . . . ,kqn be the indices in K(n). The Vandermonde determinant Qn(z1, . . . , zqn) := det(zj i )n i,j=1 Work supported in part by NSF grant DMS-0300784. R. Rumely (B) Department of Mathematics, University of Georgia, Athens, GA 30602, USA e-mail: [email protected]