Explicit constructions for sequences achieving equidistribution on capacity-one polydisks and ball

Construct explicit sequences {F_n} of generic polynomial mappings with integer coefficients of the form F_n(z1, z2) = (F1(z1, z2) + a1, F2(z1, z2) + a2), where F1 and F2 are homogeneous of the same degree, such that deg(F_n) → ∞ and the height H_E(F_n) relative to the compact, circled pseudoconvex set E tends to 0, for the two classes of capacity-one sets: (i) polydisks E = { |z1| ≤ r1, |z2| ≤ r2 } with r1 r2 = 1, and (ii) the Euclidean ball E = { ||z|| ≤ e^{1/4} }.

Background

The paper proves an equidistribution theorem (Theorem 1.4) for zeros of generic polynomial mappings in C^2 with integer coefficients, relative to compact, circled sets E in C^2 with homogeneous capacity cap_h(E)=1. The result asserts that, under conditions including deg(F_n)→∞ and small height H_E(F_n), normalized zero measures converge to the Monge–Ampère measure μ_E.

For the unit polydisk, explicit examples of sequences {F_n} satisfying these hypotheses are given (e.g., F_n(z1, z2) = (z1^n − 1, z2 − 1)). The authors note that the homogeneous capacity of a polydisk { |z1| ≤ r1 } × { |z2| ≤ r2 } is r1 r2, and for the Euclidean ball of radius r centered at the origin it is r^2 e^{-1/2}; thus cap_h(E)=1 requires r1 r2=1 for polydisks and r=e^{1/4} for the ball.

Despite these capacity characterizations, the authors explicitly state that it is unclear how to construct analogous explicit sequences {F_n} for polydisks with r1 r2=1 beyond the unit case, or for the Euclidean ball of radius e^{1/4}. This identifies a concrete gap between existence (guaranteed by general theory) and explicit constructions.