$\Theta$-Hilbertianity and strong $\Theta$-Hilbertianity

TL;DR

This paper investigates the relationship between two generalizations of Hilbertianity, showing that for certain fields like PAC fields, the notions of $p$-Hilbertianity and strong $p$-Hilbertianity are equivalent.

Contribution

It proves that for PRC and PAC fields, the concepts of $p$-Hilbertianity and strong $p$-Hilbertianity coincide, addressing a question posed by Jarden.

Findings

$p$-Hilbertianity and strong $p$-Hilbertianity are equivalent for PRC fields.

The results extend to PAC fields, confirming the conjecture in these cases.

Provides a broader understanding of Hilbertianity generalizations.

Abstract

$\Theta$-Hilbertianity and its strengthening, strong $\Theta$-Hilbertianity, are two generalizations of Hilbertianity inspired by Jarden's definition of $p$-Hilbertianity and strong $p$-Hilbertianity. Jarden has asked whether the two notions defined by him are actually the same. We address this question in its more general version of $\Theta$-Hilbertianity and show that for PRC, and, in particular, for PAC fields, $p$-Hilbertianity and strong $p$-Hilbertianity coincide.