Polynomials defining distinguished varieties in a generalised version of symmetrised bidisc

Abstract

In this study, we explored the characterisation of algebraic curves known as distinguished varieties that exhibit special boundary behavior of exiting the symmetrised bidisc exclusively through its distinguished boundary. Let 𝔻 be the open unit disc, 𝕋 be the unit circle, and 𝔼 be the ℂ\𝔻 in ℂ. Let 𝔻2 = 𝔻 ×𝔻 be the unit bidisc in ℂ2. For a polynomial 𝑝(𝑧,𝑤) ∈ ℂ[𝑧,𝑤] such that its zero set 𝑍(𝑝) ⊆ 𝔻2 ∪𝕋2 ∪𝔼2, 𝑍(𝑝)∩𝔻2 is a distinguished variety in 𝔻2. Let the symmetrisation map 𝜋:ℂ2 → ℂ2 be defined by 𝜋(𝑧,𝑤) = (𝑧 +𝑤,𝑧𝑤), and define the symmetrised bidisc 𝔾 = 𝜋(𝔻2), the distinguished boundary of 𝔾 be 𝑏Γ = 𝜋(𝕋2), and the exterior of 𝔾 be Ω = 𝜋(𝔼2). For a polynomial 𝑞(𝑠,𝑝) ∈ ℂ[𝑠,𝑝] such that its zero set 𝑍(𝑞) ⊆ 𝔾 ∪ 𝑏Γ ∪Ω, the set 𝑍(𝑞) ∩𝔾 is a distinguished variety in 𝔾. It is proven that 𝑊 ⊂ 𝔾 is a distinguished variety in 𝔾 if and only if there exists a distinguished variety 𝑉 in 𝔻2 such that 𝑊 = 𝜋(𝑉). In this study, we partially generalised this result by considering a generalised version of symmetrised bidisc. Considering the map 𝜋̃: ℂ2 → ℂ2 given by 𝜋̃(𝑧,𝑤) = (𝑧 + 𝑤,𝑧2 + 𝑤2), let 𝔾 ̃ = 𝜋̃(𝔻2). By defining a distinguished variety in 𝔾 ̃ in a similar fashion, we proved that 𝑊 distinguished variety in 𝔾 that 𝑊 ̃ ⊂𝔾 ̃ is a ̃ if and only if there exists a distinguished variety 𝑉 in 𝔻2 such ̃ =𝜋̃(𝑉).