On rank in algebraic closure

• Amichai Lampert ^[1] ; Tamar Ziegler ^[2]
  1. [1] University of Michigan–Ann Arbor

     University of Michigan–Ann Arbor

     City of Ann Arbor, Estados Unidos

     
  2. [2] Hebrew University of Jerusalem

     Hebrew University of Jerusalem

     Israel

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 1, 2024, 19 págs.
• Idioma: inglés
• DOI: 10.1007/s00029-023-00903-5
• Enlaces
  • Texto completo
• Resumen

  • 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.

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