Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals

Abstract

Let m,n≥3, (m−1)(n−1)+2≤p≤mn, and u=mn−p. The set Ru×n×m of all real tensors with size u×n×m is one to one corresponding to the set of bilinear maps Rm×Rn→Ru. We show that Rm×n×p has plural typical ranks p and p+1 if and only if there exists a nonsingular bilinear map Rm×Rn→Ru. We show that there is a dense open subset O of Ru×n×m such that for any Y∈O, the ideal of maximal minors of a matrix defined by Y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset T of Rn×p×m and continuous surjective open maps ν:O→Ru×p and σ:T→Ru×p, where Ru×p is the set of u×p matrices with entries in R, such that if ν(Y)=σ(T), then rankT=p if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal.