A Method to Compute Unitaries Representing Reducible Minimal Inner Toral Polynomials using Direct Sum

Abstract

An inner toral polynomial is a polynomial in two complex variables, p(z,w)∈C[z,w],such that its zero set is contained in〖 D〗^2∪T^2∪E^2, where D, T, and E are the open unit disk, unit circle and exterior of the closed unit disk, respectively. We say the bidegree of p(z,w)∈C[z,w] is (n,m) if p has degree n in z and m in w. An inner toral polynomial p is called a minimal inner toral polynomial if it divides any other polynomial with the same zero set as itself. In the paper Agler, J. & McCarthy, J. (2005) Distinguished Varieties. Acta Math 194, 133-153, the authors proved the existence of unitary matrices representing inner toral polynomials. Specifically, given a minimal inner toral polynomial p(z,w) of bidegree (n,m), there exists a unitary matrix, written in block form as (■(A&B@C&D)), such that det⁡(■(A-wI_m&zB@C&zD-I_n )) is a constant multiple of p(z,w). Here, blocks A,B,C and D are matrices with complex entries and of sizes (m×m),(m×n),(n×m) and (n×n) respectively. We call such unitary matrices unitaries representing p. In this work we focused on constructing a method to compute unitaries representing reducible minimal inner toral polynomials using unitaries representing its factors. We prove that if the minimal inner toral polynomial p is a product of s distinct irreducible factors, say, p_1,p_2,…,p_s, and if U_k=(■(A_k&B_k@C_k&D_k )) is a unitary representing p_k for k=1,2,3,…,s, then the matrix U=(■(⨁_D▒A_k &⨁_D▒B_k @⨁_D▒C_k &⨁_D▒D_k )) is a unitary representing p, where ⨁_D▒A_k =(■(■(A_1&0@0&A_2 )&■(…&0@0&⋮)@■(⋮&0@0&0)&■(⋱&0@0&A_s ))), the diagonal-wise direct sum of the block matrices A_ks and ⨁_D▒B_k ,⨁_D▒C_k and ⨁_D▒D_k are defined in similar fashion.