Resumen de On rank in algebraic closure

Amichai Lampert, Tamar Ziegler

• Let k be a field and Q ∈ k[x1,..., xs] a form (homogeneous polynomial) of degree d > 1. The k-Schmidt rank rkk(Q) of Q is the minimal r such that Q = r i=1 Ri Si 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 rkk(Q) in terms of rkk¯(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 rkk(Q) is large.