A Nullstellensatz for the field of complex Laurent series
Abstract
Let T be the theory of the valued field of complex Laurent series ℂ( (t) ) in the language of rings expanded by a constant for t and by predicates for n-th powers. We introduce a t-adic analogue of Kochen's operator, namely γ(x)=x/(1+tx2), and the associated Kochen ring ΛL of a field extension L of a model K of T. Our main result characterizes the field extensions of K which embed into a model of T (the formally T-adic} fields): L is formally T-adic if and only if 1/t∉ΛL. As an application we prove a Nullstellensatz for ℂ( (t) ): a polynomial f∈ K[X1,…,Xn] vanishes on the zero set of an ideal I if and only if fm=∑iωifi for some mgeqslant 1, generators fi of I and coefficients ωi in the ring ΛK(X)· K[X]. We derive the resulting correspondence between R-ideals and algebraic sets, and we compare natural incarnations of the associated T-spectrum: prime ideals, morphisms into models of T, and the space of types of T.
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BibTeX
@misc{kovacsics2026nullstellensatz,
title = {{A Nullstellensatz for the field of complex Laurent series}},
author = {Pablo Cubides Kovacsics and Sebastián Uribe Rincón},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14417},
eprint = {2610.14417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14417},
}
RIS
TY - UNPB
TI - A Nullstellensatz for the field of complex Laurent series
AU - Pablo Cubides Kovacsics
AU - Sebastián Uribe Rincón
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14417
UR - https://arxiv.org/abs/2610.14417
AB - Let T be the theory of the valued field of complex Laurent series ℂ( (t) ) in the language of rings expanded by a constant for t and by predicates for n-th powers. We introduce a t-adic analogue of Kochen's operator, namely γ(x)=x/(1+tx^2), and the associated Kochen ring Λ_L of a field extension L of a model K of T. Our main result characterizes the field extensions of K which embed into a model of T (the formally T-adic} fields): L is formally T-adic if and only if 1/t∉Λ_L. As an application we prove a Nullstellensatz for ℂ( (t) ): a polynomial f∈ K[X_1,…,X_n] vanishes on the zero set of an ideal I if and only if f^m=∑_iω_if_i for some mgeqslant 1, generators f_i of I and coefficients ω_i in the ring Λ_K(X)· K[X]. We derive the resulting correspondence between R-ideals and algebraic sets, and we compare natural incarnations of the associated T-spectrum: prime ideals, morphisms into models of T, and the space of types of T.
ER -