• Corpus ID: 251134883

Isogeny classes of cubic spaces

@inproceedings{Bik2022IsogenyCO,
  title={Isogeny classes of cubic spaces},
  author={Arthur Bik and Alessandro Danelon and Andrew Snowden},
  year={2022}
}

• A. Bik, Alessandro Danelon, A. Snowden
• Published 28 July 2022
• Mathematics

. A cubic space is a vector space equipped with a symmetric trilinear form. Two cubic spaces are isogeneous if each embeds into the other. A cubic space is non-degenerate if its form cannot be expressed as a finite sum of products of linear and quadratic forms. We classify non-degenerate cubic spaces of countable dimension up to isogeny: the isogeny classes are completely determined by an invariant we call the residual rank , which takes values in N ∪ {∞} . In particular, the set of classes is discrete and (under the partial order of embedability) satisfies the descending chain condition. each embeds into the other 1 . We define the strength of f ∈ P n ( V ), denoted str( f ), to be the infimum of those s for which there is an expression f = P si =1 g i h i where g i and h i are homogeneous elements of the ring P ( V ) = L j ≥ 0 P j ( V ) of smaller degree than f ; if there is no such expression for any s then str( f ) = ∞ . We say that an n -space is non-degenerate if its defining form has infinite strength.