Generalization of the symmetricity properties of polynomials defining distinguished varieties on the open unit bidisk

Abstract

Let 𝔻 be the open unit disk, 𝕋 be the unit circle, and 𝔼 be the exterior of the closed unit disk in ℂ. A polynomial p(z, w) is said to define a distinguished variety on 𝔻², if Z(p) = { (z, w) ∊ ℂ²∶ p(z, w) = 0 } ⊆ 𝔻²∪𝕋² ∪ 𝔼². For such polynomials, the zero set Z(p) inside 𝔻²is called a distinguished variety on 𝔻². For a polynomial p(z, w) with two variables having bidegree (n, m), the reflection is defined by p̃(z, w) = zⁿwᵐp (Mathematical Formula)). A polynomial p ∊ ℂ[z, w] is essentially 𝕋² −symmetric if p(z, w) = c p̃(z, w) for some c ∊ 𝕋. In 2010, Greg Knese introduced the concept of symmetricity for polynomials defining distinguished varieties on 𝔻2 and has shown that a polynomial p defining a distinguished variety on 𝔻² is essentially 𝕋² −symmetric. In this study, this concept of symmetricity is generalized for polynomials defining distinguished varieties on open unit polydisk 𝔻ⁿ, by considering polynomials with n variables. A polynomial p(z₁, z₂, ... , z(n)) is said to define a distinguished variety on 𝔻ⁿ, if Z(p) = { (z₁, z₂, ... ,z(n)) ∊ ℂⁿ∶ p(z₁, z₂, ... , z(n)) = 0 } ⊆𝔻ⁿ ∪ 𝕋ⁿ ∪ 𝔼ⁿ. For such polynomials, the zero set Z(p) inside 𝔻ⁿ is called a distinguished variety on 𝔻ⁿ. For a polynomial p(z₁, z₂, ... , z(n) with n variables having degree (m₁, m₂, ... , mn), the reflection is introduced by p̃(z₁, z₂, ... , z(n)) =z₁ᵐ¹z₂ᵐ² ... z(n)ᵐⁿp (Mathematical Formula)). We defined a polynomial p ∊ ℂ[z₁, z₂, ... , z(n)] to be essentially 𝕋ⁿ −symmetric if p(z₁, z₂, ... , z(n)) = cp̃(z₁, z₂, ... , z(n) ) for some c ∊ 𝕋. This study proves that a polynomial p defining a distinguished variety on 𝔻ⁿ is essentially 𝕋ⁿ −symmetric for any 2 ≤ n < ∞. Future studies can focus on proving properties that already exist for two variable polynomials in the case of polynomials with n variables.