On rank in algebraic closure

Amichai Lampert*, Tamar Ziegler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let k be a field and Q∈ k[x1, … , xs] a form (homogeneous polynomial) of degree d> 1. The k -Schmidt rank rk k(Q) of Q is the minimal r such that Q=∑i=1rRiSi with Ri, Si∈ k[x1, … , xs] forms of degree < d . When k is algebraically closed and char (k) doesn’t divide d, this rank is closely related to codimAs(∇Q(x)=0) - also known as the Birch rank of Q. When k is a number field, a finite field or a function field, we give polynomial bounds for rk k(Q) in terms of rk (Q) where k¯ is the algebraic closure of k. Prior to this work no such bound (even ineffective) was known for d> 4 . This result has immediate consequences for counting integer points (when k is a number field) or prime points (when k= Q) of the variety (Q= 0) assuming rk k(Q) is large.

Original languageAmerican English
Article number15
JournalSelecta Mathematica, New Series
Volume30
Issue number1
DOIs
StatePublished - Feb 2024

Bibliographical note

Publisher Copyright:
© 2024, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

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