×

Non-existence of the Artin function for Henselian pairs. (English) Zbl 0803.13005

If \((A,m)\) is a local Henselian ring, the strong version of the Artin approximation theorem says that given a system of algebraic equations over \(A\), a sufficiently good approximate solution in the \(m\)-adic topology can be approximated by an exact solution. The Artin function is a quantitative measure of “sufficiently good” in the above statement. The main point of this paper is to give a simple counterexample to the analogue of the above statement for Henselian pairs \((A,I)\).

MSC:

13B40 √Čtale and flat extensions; Henselization; Artin approximation
13J15 Henselian rings
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Ogoma, T.: General N?ron desingularization based on the idea of Popescu. Preprint · Zbl 0821.13003
[2] Pfister, G., Popescu, D.: Die strenge Approximationseigenschaft lokaler Ringe. Invent. Math.30, 301-307 (1975) · Zbl 0308.13026
[3] Popescu, D.: General N?ron desingularization. Nagoya Math. J.100, 97-126 (1985) · Zbl 0561.14008
[4] Popescu, D.: General N?ron desingularization and approximation. Nagoya Math. J.104, 85-115 (1986) · Zbl 0592.14014
[5] Spivakovsky, M.: Smoothing of ring homomorphisms, aproximation theorems and the Bass-Quillen conjecture. Preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.