# Analytic Functions over Fields other than Real or Complex Numbers

Let K denote either the field of real numbers or the complex fields. An analytic function over $K^n$ is a function that can be represented locally by a convergent power series in n variables with coefficients in K.

My question is that can we take K to be other fields? It seems that such a field K should satisfy some criteria:

1. It is a metric space or at least a topological space.
2. There should be complete, in the sense that Cauchy nets converge.
3. The above 2 points probably force that K should have cardinality at least the size of the continuum.

Can there be other K where a reasonable theory of analytic functions can be developed? Say for cardinalities larger than the continuum? Probably this invokes some model theory.

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I believe there is a pretty rich theory of p-adic functions. – Steve Huntsman Apr 27 '10 at 16:11
You might also want to look at Serre's book "Lie groups and Lie algebras" (or something like this). He does the basic theory of Lie groups with respect to a complete absolute-valued field using "analytic" in place of "smooth" (this is acceptable by that old theorem of Montgomery-Zippin-???). Included is a correspondence between Lie algebras, formal group laws and "group chunks", which are like a stand-alone neighborhood on which the formal group 'tries its best' to be an actual group law. One of the central ideas is the use of the Baker-Campbell-Hausdorff formula in this general situation. – Sean Rostami Apr 27 '10 at 16:47
Just wanted to say that the continuum might have any regular uncountable cardinality you want. This is a result by Easton. So the size of $K$ isn't the point – Stefan Hoffelner Apr 27 '10 at 17:08
See the many answers. But note: locally given by power series is NOT what you want. There are many strange functions on the $p$-adics that are CONSTANT in some neighborhood of every point, but lacking connectedness one cannot conclude the function is constant. – Gerald Edgar Apr 27 '10 at 17:19
@Gerald: sometimes it is and sometimes it isn't. E.g., that definition of analytic function is good enough to do Lie theory: see Serre's Lie Algebras and Lie Groups. For geometric applications, yes, it's often better to have a more rigid collection of analytic functions: it depends on what you're trying to do. – Pete L. Clark Jun 1 '10 at 19:04

There are several rich theories of analysis on non-archimedian theories. Neal Koblitz' book on $p$-adic analysis is a good introduction. Non-archimedian analysis by Bosch, Güntzer and Remmert is more encyclopedic. Berkovich's Spectral Theory and Analysis over Non-archimedian Fields introduces his beautiful theory of analytic spaces allowing for a reasonable algebraic topological theory. In Goss's book Basic Structures of Function Field Arithmetic there is a good introduction to analysis in positive characteristic.

Your suggestion that this subject might have something to do with model theory is apt. As the above references show, the theory may be developed without model theory, but it has been studied intensively via model theory giving interesting results about quantifier elimination, uniformity across the $p$-adics, and establishing a basis for motivic integration. You might want to look at the paper by van den Dries and Denef, $p$-adic and real subanalytic sets. Ann. of Math. (2) 128 (1988), no. 1, 79--183.

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But p-adics have zero characteristic, so why to call this "analysis in positive characteristic"? (or perhaps i just misread your answer...) – Qfwfq Apr 27 '10 at 17:39
My point was that analysis may be developed over (complete -- though even this restriction may be relaxed) valued fields of arbitrary characteristic. In Goss's book, the ground field is a completion of the algebraic closure of a field of formal Laurent series over a field of positive characteristic. Most of my other suggested references focus on p-adic fields, but even they allow for more general fields. – Thomas Scanlon Apr 27 '10 at 17:53

There is a perfectly working theory of analytic functions over the p-adics with lots of theorems. No model theory is needed (Neal Koblitz has a book about that, also Non-Archimedean Analysis by Bosch, Güntzer, Remmert for a dry treatise), but indeed we face (ultra)metric complete fields here, being uncountably infinite, just as you suggest.

Nonetheless, if you just need the "feel" of power series to model something abstractly, formal power series, cf. http://en.wikipedia.org/wiki/Formal_power_series, may be all you need. They behave in many ways like (convergent) power series, for example if you 'formally' wish to invert a differential operator, such computations - at least algebraically - may be given a more-or-less solid foundation in a formal power series ring.

All classical operations, e.g. taking derivatives etc, can be defined termwise, no problem. You can also plug formal power series into each other, but just if the constant coefficient is zero, sadly.

Finally, your two points do not really enforce large cardinality. A finite field can be equipped with the discrete metric, this makes it complete, so you could take about convergent power series over this - it just means that only finitely many coefficients can be non-zero, making it effectively a polynomial ring.

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I was looking for something analytic. Wasn't really thinking about formal power series. – Colin Tan Apr 28 '10 at 14:26

If you don´t mind skew fields, have a look at quaternionic analysis. Interestingly, many of the facts of complex analysis don´t hold there at all! (In fact, all you need to do analysis, is a Banach algebra or a complete ring with non-trivial unitary group.)

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Unfortunately, the space of quaternionic-analytic functions of one quaternionic variable just happens to coincide with the space of 4-tuples of real-analytic functions of 4 real variables. – Qfwfq Apr 27 '10 at 17:41
This is far from satisfactory, but: Convergent power series with real coefficients make sense when interpreted as functions on the quaternions. This allows one to define a lot of old favorites like the exponential function on the quaternions. It's challenging, however, to determine the appropriate notion of "Riemann surface" for some of these, like the logarithm, since it has uncountably many branches. – Daniel Asimov Jun 1 '10 at 20:11