Projective Nullstellensatz for not necessarily algebraically closed fields
Rati Ludhani
TL;DR
The paper extends Hilbert’s Nullstellensatz to projective settings over not-necessarily algebraically closed fields, with a new, computationally efficient projective analogue for finite fields. Over infinite fields, the projective version follows from the affine Hilbert $K$-Nullstellensatz, yielding $\\mathsf{I}(\\mathsf{V}_K(I))=\\sqrt[K]{I}$ when nonempty and $S$ when empty. For finite fields, the authors prove a projective $\\mathbb{F}_q$-Nullstellensatz using a simple colon-ideal $$(I+\\Gamma_q^*(k)):\\frak{d}$$ with $d=(d_1+\cdots+d_r)(q-1)+1$, where $\\frak{d}=\langle X_0^d,\ldots,X_n^d\rangle$, extending to algebraic extensions via $s$ and $\\frak{s}_F$. They also provide counterexamples to three of Laksov–Westin’s conjectures on Nullstellensatz strengthening, clarifying limitations of certain proposed universal forms and highlighting the improved efficiency of their finite-field projective form.
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-José 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.