Abstract
In this survey we determine an explicit set of generators of the maximal ideals in the ring \(\mathbb{R}[x_{1},\ldots,x_{n}]\) of polynomials in n variables with real coefficients and give an easy analytic proof of the Bass-Vasershtein theorem on the Bass stable rank of \(\mathbb{R}[x_{1},\ldots,x_{n}]\). The ingredients of the proof stem from different publications by Coquand, Lombardi, Estes and Ohm. We conclude with a calculation of the topological stable rank of \(\mathbb{R}[x_{1},\ldots,x_{n}]\), which seems to be unknown so far.
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Notes
- 1.
For example in the case j = 2 and n = 3, \(x_{1}x_{2}\mapsto 0\), \(x_{1}x_{2}x_{3}\mapsto 0\) and \(x_{2} + x_{3}\mapsto x_{3}\).
- 2.
For later purposes we note that, by the same reason, if I and P are ideals, P prime and \(I \subseteq P\), then there exist minimal prime ideals P min with \(I \subseteq P_{min} \subseteq P\).
- 3.
If i = 1, then we consider IR (a 0).
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Mortini, R., Rupp, R. (2014). The Ring of Real-Valued Multivariate Polynomials: An Analyst’s Perspective. In: Douglas, R., Krantz, S., Sawyer, E., Treil, S., Wick, B. (eds) The Corona Problem. Fields Institute Communications, vol 72. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1255-1_8
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