28.11.2012 23:47, quasi wrote: > Let m,n be positive integers, and let k be an integer with > 0 <= k <= min(m,n). The set T_k of m x n matrices of > rank <= k is easily seen to be closed since, for each k, > there is a polynomial P_k in m*n variables with real > coefficients such that an m x n matrix A with real entries > satisfies the condition rank(A) <= k iff the coefficients > of A satisfy P_k = 0. Regarding P_k as a function from > R^(mxn) to R, P_k is continuous, hence ((P_k)^(-1))(0) > is closed. It follows that T_k is closed for all k. In > particular, for each k, T_(k-1) is closed, and thus, > the set of matrices with rank >= k is open.
Sounds good. But how to prove the existence of the polynomials P_k?