An efficient resultant for determining reciprocal zeros in polynomials

We create a new resultant for determining the presence and number of reciprocal zeros in a given degree-n polynomial a(z) whose coefficients are real numbers. While the 2n × 2n Sylvester resultant (or eliminant) could be used for this purpose, our new resultant is based on a simple (n + 1) × (n + 1) matrix. The number of reciprocal zeros present in a(z) can then be determined from the rank of this matrix. It extends to matrix form some of the work described by Muir in 1897 for determinants. Muir's work is also shown to lead to a simple factorization of the Bézout resultant for the same problem.