Rank restricting functions (Q1410725)

The authors determine real-valued functions \(f\) such that whenever \((a_{ij})\) has rank at most \(k\), \((f(a_{ij}))\) has rank at most \(d\). These functions for \(k\geq 2\) are polynomials the degrees of whose terms are small compared with \(d\), which is typically considerably larger than \(k\); for \(k=1\) the class is given as elementary functions satisfying a differential equation. Similar results are proved for complex functions.