A generalization of the converse of Brolin’s theorem

Abstract

We prove a generalization of Lopes's theorem, that is, of the converse of Brolin's theorem.

Irreducibility of lemniscates

“…Since S is a polynomial, (13) implies that δ also is a polynomial. Moreover, deg δ = 1 (because δ is a Möbius transformation), thus (13) contradicts (11). The obtained contradiction shows that P does not satisfy the Composition Condition.…”

“…Notice that if P is a polynomial, the composition condition reduces to the condition that P = W k for some polynomial W and k ≥ 2. Its necessity for the reducibility of L P (x, y) in the polynomial case was established by the first author in [12] (notice that this result has found applications to complex dynamics, see [11]). In Section 3.2, we show however that for rational P the reducibility of L P (x, y) = 0 does not imply in general the composition condition.…”