Absolute constants of Koras-Russell-liked threefolds

TL;DR

This paper investigates whether the Absolute Constants ring $\mathrm{AK}(R_{F,\eta}/F)$ attached to Koras-Russell-type threefolds remains invariant under translates of invariant data. By developing a framework with exponential maps, locally finite higher derivations, and degree filtrations $\mathcal{F}_1$, $\mathcal{F}_2$ together with gradings $\omega_1$, $\omega_2$, the authors analyze translates of the defining equations $X^2Y+Z^2+T^3+\eta(X)$ and related rings $R_{\eta_c}$, $S_c$, and $S_0$. They prove that for all $\eta$, the absolute constants satisfy $\mathrm{AK}(R_{F,\eta})=F[x]$, implying invariance under translates, and they explicitly connect these results to the case $\eta=0$. The findings suggest a possible broader property of Absolute Constants that could facilitate calculations on other affine curves and motivate questions about the generality of this invariance across wider classes of varieties.

Abstract

This result generalizes a previous result established in [2] where the Absolute Constants of the Koras-Russell threefold was shown to be invariant under translates in the base field to the Absolute Constants of the Koras-Russell threefold being invariant under translates of any of its absolute invariants. It must be pointed out that there is no prior reason for the ring of Absolute Constants to be invariant under so general translates, but remains a possibility that it is part of a more general property of the Absolute Constants ring.

Theorems & Definitions (10)

• Remark 1
• proof
• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof