The first step in lower bounding the border rank of a tensor or polynomial
with border apolarity is to enumerate all ideals contained in the
annihilator with Hilbert series equal to the Hilbert series of an ideal of
general points. The second step requires determining whether any such ideal
may be deformed to an ideal of points. Typically, one simplifies these
questions by asking if there are any such ideals which are additionally
fixed under a given solvable group of symmetries of the tensor or
polynomial.
In this talk I discuss the challenges involved in the ideal enumeration
step. At a high level, the ideals are enumerated multigraded component by
component, but concrete questions arise. How should partially constructed
ideals be represented? How are the symmetries of the tensor or polynomial
handled? How do we proceed when the answer contains positive dimensional
families? Furthermore, I anticipate the successful application of both
steps of border apolarity will as much as possible interleave checks for
deformability of partially built ideals into the early steps of
enumeration. I hope this discussion will make clear the context in which
tests for deformability will need to be applied.