Real Nullstellensatz in RCA_0: Formalization and Provability

Develop a reverse-mathematical coding of the Real Nullstellensatz (for a real closed field F and a prime ideal I in F[x1,…,xn], the variety VF(I) is nonempty if and only if whenever p1,…,pm ∈ F[x1,…,xn] and ∑i pi^2 ∈ I, then each pi ∈ I) within the language of second-order arithmetic used by RCA_0, and determine whether the resulting formalization is provable over RCA_0.

Background

The paper surveys reverse-mathematical strengths of algebraic and real-closed-field results, showing that various Nullstellensatz formulations for algebraically closed fields are provable in RCA_0, while stronger algebraic principles (e.g., Hilbert’s Basis Theorem in general) require more induction. In the real-closed setting, standard correspondences from the algebraically closed case fail over the reals, but a Real Nullstellensatz holds for real closed fields.

The authors note prior results on real closures in RCA_0, WKL_0, and ACA_0, and then highlight a gap: although the Real Nullstellensatz is known in ZFC, its status in RCA_0 has not been settled. Moreover, there is a foundational obstacle—the appropriate formalization of the Real Nullstellensatz within RCA_0 is itself unclear—making both the coding and the provability open issues in the reverse mathematics framework.