I am searching for good references on the topic of the behaviour of the elements in the algebraic closed field $(\mathbb{R}[x_{1},\dots,x_{n}])^{\operatorname{alg}}.$ I imagine that, when we try to see these as partial multivalued functions from $A\subseteq\mathbb{C}^{n}\to\mathbb{C},$ they are very wild on $(x_{1},\dots,x_{n})$ but I suppose that some of their properties should be known and further studied. I am interested in introductory books, papers or advanced monographs describing these kind of fields and their geometric properties.
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2$\begingroup$ What makes you think that an element of this field can be viewed as a function on $\mathbb{C}^n$? $\endgroup$– Alex KruckmanCommented Mar 12, 2021 at 19:53
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1$\begingroup$ For $n=1$ they are the Puisseaux series that are algebraic over $\mathbb{C}(x)$. I don't see why they could be defined as functions on $\mathbb{C}$. May be only on some small balls. $\endgroup$– NulhomologousCommented Mar 12, 2021 at 20:48
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$\begingroup$ @AlexKruckman Yeah, sorry, I meant partial functions defined on some $A \subseteq \mathbb{C}^{n}$ as it is clear that it is not even true that rational functions are defined on all of $\mathbb{C}^{n}.$ However, now I see that these are in fact multivalued functions in general. I am precisely asking for references to understand this field better. I am going to correct this. Thank you. $\endgroup$– HvjurthukCommented Mar 12, 2021 at 21:09
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$\begingroup$ Most of the time a field like this comes up, one just wants to understand it using algebra - as an algebraically closed field, it shares most important properties with other algebraically closed fields, including the easier-to-visualize $\mathbb C$. I know of fewer cases where it's helpful to think about the elements as functions. It might (I'm not sure) help to be more specific about what kinds of questions about this field you what references to help answer. $\endgroup$– Will SawinCommented Mar 12, 2021 at 23:52
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2$\begingroup$ If you only work with finitely many elements of the field at a time, there will exist an $n$-dimensional algebraic variety mapping to $\mathbb C^n$ on which all will be well-defined functions. You can view them as functions on this variety and look at their zero set there. $\endgroup$– Will SawinCommented Mar 13, 2021 at 0:59
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