Jayasingha, J.A.T.D.Wijesooriya, U.D.2025-11-062025-11-062025-11-07Proceedings of the Postgraduate Institute of Science Research Congress (RESCON)-2025, University of Peradeniya P-63ISSN3051-4622https://ir.lib.pdn.ac.lk/handle/20.500.14444/6130In 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).enDistinguished varietyInner toral polynomialsSymmetrisation mapSymmetrised bidiscArticle