Projective Nullstellensatz for not necessarily algebraically closed fields

TL;DR

This paper extends the Nullstellensatz to projective varieties over fields that are not necessarily algebraically closed, introducing a more efficient set and providing counterexamples to existing conjectures.

Contribution

It introduces a more efficient set for the projective Nullstellensatz over arbitrary fields and disproves three conjectures by Laksov and Westin.

Findings

Established a more efficient set for the projective Nullstellensatz over finite fields.

Provided counterexamples to three conjectures of Laksov and Westin.

Extended Nullstellensatz applicability beyond algebraically closed fields.

Abstract

The Nullstellensatz, proved by Hilbert in 1893, is a classical result that holds when the base field is algebraically closed. When the base field is finite, a version of Hilbert's Nullstellensatz is given by Terjanian in 1966. Laksov in 1987 generalized Hilbert's Nullstellensatz to a $K$-Nullstellensatz when the base field $K$ is not necessarily algebraically closed. However, unlike Tarjanian's Nullstellensatz, Laksov's Nullstellensatz is not very explicit. Later, Laksov and Westin in 1990 proposed a strengthening to Laksov's Nullstellensatz in the form of four conjectures. A projective analogue of Nullstellensatz of the classical Nullstellensatz of Hilbert is well-known for projective varieties over algebraically closed fields. For finite fields, the projective analogue of the Nullstellensatz can be derived as an application of Hilbert's Nullstellensatz, though it is not as efficient as Terjanian's Nullstellensatz. Gimenez, Ruano and San-Jos\'e in 2023 strengthened this result by identifying a computationally efficient set. Here, we introduce an even more efficient set that establishes the projective analogue of the Nullstellensatz for finite fields. Additionally, we provide counterexamples to three of the conjectures of Laksov and Westin.