On the construction of unitaries representing minimal inner toral polynomials

A polynomial p=p(z,w)∈C[z,w] is called an inner toral polynomial if the zero set of p, contained in D^2∪T^2∪ E^2 where D^2 is the open bidisk, T^2 is the two dimensional torus and E^2 is the exterior of the closed bidisk in C^2. Such a polynomial is called a minimal inner toral polynomial if it divides all the other polynomials with same zero set as itself. The bidegree of p is defined as the ordered pair (n,m) whenever p has degrees n and m in variables z and w respectively. For a minimal inner toral polynomial p(z,w) of bidegree (n,m), there exist block unitary matrices, (■(A&B@C&D))_(((m+n)×(m+n)) ), such that p(z,w) is a constant multiple of the determinant of (■(A-wI_m&zB@C&zD-I_n )), where, blocks A,B,C and D are matrices with complex entries of sizes (m×m),(m×n),(n×m) and (n×n) respectively. Because the size of such unitaries solely depend on the bidegree of the considered minimal inner toral polynomial, the process of computing such unitary matrices is manageable when the bidegree is very small but become strenuous exponentially with the bidegree. In this work, we introduce a method of constructing unitary matrices representing reducible minimal inner toral polynomials in terms of the unitaries representing its factors and vice versa.