## Abstract

Let k be a field and Q∈ k[x_{1}, … , x_{s}] 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 R_{i}, S_{i}∈ k[x_{1}, … , x_{s}] 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 _{k¯}(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 language | English |
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Article number | 15 |

Journal | Selecta Mathematica, New Series |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2024 |

### Bibliographical note

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