No short polynomials vanish on bounded rank matrices
Abstract
We show that the shortest nonzero polynomials vanishing on boundedrank matrices and skewsymmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$minors or $t$Pfaffians of a generic matrix or skewsymmetric matrix one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a sufficiently general $t$dimensional subspace of an affine $n$space does not contain polynomials with fewer than $t+1$ terms.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.11764
 Bibcode:
 2021arXiv211211764D
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 14M12;
 14Q20;
 15A15
 EPrint:
 13 pages, comments welcome