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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nash rings on planar domains


Author: Gustave A. Efroymson
Journal: Trans. Amer. Math. Soc. 249 (1979), 435-445
MSC: Primary 14J05; Secondary 13F15, 14G30
MathSciNet review: 525683
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Abstract: Let D be a semialgebraic domain in $ {R^2}$. Let $ {N_D}$ denote the Nash ring of algebraic analytic functions on D. Let $ {A_D}$ denote the ring of analytic functions on D. The main theorem of this paper implies that if $ \mathcal{B}$ is a prime ideal of $ {N_D}$, then $ \mathcal{B}{A_D}$ is also prime. This result is proved by considering $ p\left( {x,\,y} \right)$ in $ \textbf{R}[{x,\,y}]$ and showing that $ p({x,\,y})$ can be put into a form so that its factorization in $ {N_D}$ is given by looking at its local factorization as a polynomial in y with coefficients which are analytic functions of x. Then for more general domains, a construction using the ``complex square root'' enables one to reduce to the case already considered.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0525683-0
PII: S 0002-9947(1979)0525683-0
Keywords: Nash ring, factorization of polynomials in Nash ring
Article copyright: © Copyright 1979 American Mathematical Society