On canonical stratifications
Akira Koriyama · Kodai Mathematical Journal · 1972
It is well-known that every compact manifold can be imbedded into a Euclidean m-space R m for some m.Furthermore Nash [7] proved that for a closed connected smooth manifold M, smoothly imbedded in R m , there is a polynomial map /: R m -*R q for some q such that M is a connected component of /~α(0).A polynomial map / is an ordered set (g lt g 2 , , g q ) of polynomial functions.On the other hand, by a simple calculation, we have the following PROPOSITION A. Every polynomial can be expressed in a form of determinant of a certain square matrix whose entries are monomials of degree 1 or 0.More precisely, for any polynomial function g: R m -+R, there is a positive integer n and an afβne imbedding ψ of R m into the space M(n, n) of all nxn real matrices such that the following diagram is commutative: M(n, n) (This was communicated to the author by T. Ishikawa).REMARK.For the given polynomial map f=(gi, -, g q ): R m ->R q , we take the positive integer n common to all g t .On account of the above facts every closed connected smooth manifold can be imbedded into M(n, n) for some n and is expressed as the intersection of the q aίϊine m-spaces ψi(R m ) and det-^O) in M(n, ή).Thus it is meaningful to study the set of zeros of det: M(n, n)->R, which is the same as the set of singular matrices, or the set of matrices with rank r<n.More generally we consider, in this paper, the set of nxm real matrices, n^m, with rank r