Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud's question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context.Comment: 21 pages; presentation improved as suggested by the referees; To appear in Journal of London Mathematical Societ

Solving Diophantine Problems Over All Residue Class Fields of a Number Field

(Statement of Responsibility) by Adam Ginensky(Thesis) Thesis (B.A.) -- New College of Florida, 1977(Electronic Access) RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE(Bibliography) Includes bibliographical references.(Source of Description) This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.(Local) Faculty Sponsor: Chae, Soo Bon

Reminiscences of Zygmund

The author recall his contact with Antoni Zygmund in 1977