Nash rings on planar domains

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.