  The GM-MDS theorem, conjectured by Dau-Song-Dong-Yuen and proved by Lovett
and Yildiz-Hassibi, shows that the generator matrices of Reed-Solomon codes can
attain every possible configuration of zeros for an MDS code. The recently
emerging theory of higher order MDS codes has connected the GM-MDS theorem to
other important properties of Reed-Solomon codes, including showing that
Reed-Solomon codes can achieve list decoding capacity, even over fields of size
linear in the message length.
  A few works have extended the GM-MDS theorem to other families of codes,
including Gabidulin and skew polynomial codes. In this paper, we generalize all
these previous results by showing that the GM-MDS theorem applies to any
\emph{polynomial code}, i.e., a code where the columns of the generator matrix
are obtained by evaluating linearly independent polynomials at different
points. We also show that the GM-MDS theorem applies to dual codes of such
polynomial codes, which is non-trivial since the dual of a polynomial code may
not be a polynomial code. More generally, we show that GM-MDS theorem also
holds for algebraic codes (and their duals) where columns of the generator
matrix are chosen to be points on some irreducible variety which is not
contained in a hyperplane through the origin. Our generalization has
applications to constructing capacity-achieving list-decodable codes as shown
in a follow-up work by Brakensiek-Dhar-Gopi-Zhang, where it is proved that
randomly punctured algebraic-geometric (AG) codes achieve list-decoding
capacity over constant-sized fields.
