Zeros of Symmetric Laurent Polynomials of Type $(BC{)}_n$ and Koornwinder-Macdonald Polynomials Specialized at $t^{k+1}{q}^{r-1}=1$

A characterization of the space of symmetric Laurent polynomials of type $(BC{)}_n$ which vanish on a certain set of submanifolds is given by using the Koornwinder-Macdonald polynomials. A similar characterization was given previously for symmetric polynomials of type $A_n$ by using the Macdonald polynomials. We use a new method which exploits the duality relation. The method simplifies a part of the proof in the $A_n$ case.