Explicit semi-algebraic polynomial characterization of tensor rank beyond rank-2

Develop explicit finite collections of polynomial equalities and inequalities that provide a concrete semi-algebraic characterization of the sets of real m × n × p tensors of fixed or bounded rank for ranks greater than 2. This should include precise polynomial descriptions that decide membership in rank ≤ r and rank = r sets for r > 2 and specify any necessary conditions such as multilinear rank, thereby extending the 2 × 2 × 2 case to general m × n × p formats.

Background

The paper shows that sets of matrices or tensors of fixed or bounded rank are semi-algebraic, meaning membership can be decided by evaluating a finite number of polynomial equations and inequalities. For 2 × 2 × 2 tensors, rank can be determined using the hyperdeterminant and multilinear rank, yielding a complete decision procedure.

However, for general m × n × p tensors, the authors note that while semi-algebraicity holds abstractly, explicit polynomial descriptions are not known for ranks beyond 2. Providing such concrete descriptions would generalize the practical rank-decision tools available in the 2 × 2 × 2 case to larger tensor formats.