The set of forms with bounded strength is not closed

Abstract

The strength of a homogeneous polynomial (or form) is the smallest length of an additive decomposition expressing it whose summands are reducible forms. Using polynomial functors, we show that the set of forms with bounded strength is not always Zariski-closed. More specifically, if the ground field is algebraically closed, we prove that the set of quartics with strength $\le 3$ is not Zariski-closed for a large number of variables.

arXiv comment9 pages, removed condition on characteristic

References in corpus (8)

• Big polynomial rings and Stillman's conjecture
• The geometry of polynomial representations
• Polynomials and tensors of bounded strength
• Topological noetherianity for cubic polynomials
• Big polynomial rings with imperfect coefficient fields
• On the strength of general polynomials
• Schmidt rank and singularities
• Linear subspaces of minimal codimension in hypersurfaces

Cited by in corpus (2)

• The geometry of polynomial representations
• On the strength of general polynomials