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I am reading Kollár's recent survey on Nash's work in algebraic geometry. I am trying to understand why the retraction $\pi:U_M\to M$ introduced in Discussion 7 is a Nash map. Kollár applies Claim 8.4 to the Lagrange multiplier system (7.7). For this, one would need to know that for any $p\in U_M$ there are finitely many tuples $(x,\lambda)\in\mathbb{R}^N\times\mathbb{R}^r$ solving this system. Can someone explain why this is so, or how to get around this property? Finiteness is plausible, since we are dealing with $N+r$ equations in $\mathbb{R}^{N+r}$, but I would like to see some formal details.

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