Relative rank and regularity

The notion of Schmidt rank/strength for a collection of m polynomials plays an important role in additive combinatorics, number theory and commutative algebra; high rank collections of polynomials are “psuedorandom”.  An arbitrary collection of polynomials is not necessarily of high rank, but via a regularity procedure is contained in an ideal generated by a huge (depending on m) high rank collection of polynomials. We describe a refined notion of rank/strength that allows for a new regularization procedure with polynomial dependence on m, while maintaining the psuedorandomess properties.