Symmetricity of polynomials defining distinguished varieties on the symmetrized bidisk

Abstract

Let 𝔻 be the unit disk, 𝕋 be the unit circle, and 𝔼 be the set ℂ∖𝔻̅ in ℂ. For a polynomial 𝑓(𝑧,𝑤)∈ ℂ[𝑧,𝑤] such that its zero set 𝑍(𝑓)⊂𝔻²∪𝕋²∪𝔼², we say 𝑍(𝑓)∩𝔻² is a distinguished variety on 𝔻² and the polynomial 𝑓(𝑧,𝑤) is called a polynomial defining a distinguished variety on 𝔻². A polynomial 𝑓(𝑧,𝑤)∈ ℂ[𝑧,𝑤] is said to have bidegree (𝑛,𝑚) if 𝑓(𝑧,𝑤) has degree 𝑛 in 𝑧 and degree 𝑚 in 𝑤. The reflection of 𝑓(𝑧,𝑤) at bidegree (𝑛,𝑚) is defined as the <equation 1>. A polynomial 𝑓(𝑧,𝑤) is called essentially 𝕋²−symmetric if 𝑓(𝑧,𝑤)= 𝑐 𝑓̃(𝑧,𝑤) for some 𝑐∈𝕋. Every polynomial that defines a distinguished variety on 𝔻² is symmetric with respect to 𝕋² in the sense that it is essentially 𝕋²−symmetric. Let 𝜋∶ℂ²→ℂ² be the symmetrization map given by 𝜋∶ (𝑧,𝑤)⟼(𝑧 + 𝑤,𝑧𝑤) and 𝔾= 𝜋(𝔻²), Γ be the boundary of 𝔾 and 𝜕𝛤 = 𝜋(𝕋²) be the distinguished boundary of 𝔾, where 𝔾 is called the symmetrized bidisk. For a polynomial 𝑔(𝑠,𝑝)∈ℂ[𝑠,𝑝] such that its zero set 𝑍(𝑔)⊂ 𝔾 ∪ 𝑏Γ ∪ ℂ \ Γ, we say 𝑍(𝑔)∩ 𝔾 is a distinguished variety on the symmetrized bidisk 𝔾, and 𝑔(𝑠,𝑝) is referred to as a polynomial defining a distinguished variety on the symmetrized bidisk 𝔾. In this work, symmetric properties of polynomial defining a distinguished variety on the symmetrized bidisk 𝔾 were studied. Given a polynomial 𝑔(𝑠,𝑝) of bidegree (𝑘,𝑙), the reflection of 𝑔 was defined as <equation 2>. A polynomial 𝑔 is essentially 𝜕𝛤-symmetric if 𝑔(𝑠,𝑝)=𝑐 𝑔̂(𝑠,𝑝), where 𝑐∈𝕋. It was proved that a polynomial defining a distinguished variety on the symmetrized bidisk is essentially 𝜕𝛤- symmetric. This study contributes to the broader understanding of geometric representation and properties of polynomial defining distinguished varieties on the symmetrized bidisk.