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Stream: deprecated: mathematics

Topic: closed form expressions of functions

Ben Sprott (Jun 26 2021 at 23:49):

I am trying to develop a theory for closed form expressions for functions. I am using my little knowledge of maps of the type $f = \mathbb{R} \rightarrow \mathbb{R}$. My first thought is that expressions of this type are basically compositions of other functions under (monoidal?) products times and plus. This suggests that a theory of expressions for functions is about homsets that are enriched over monoidal categories with many monoidal products.

There is another part to this where we can kind of see that the functions of type $f = \mathbb{R} \rightarrow \mathbb{R}$ can be divided into those that have an expression and those that don't, and then further those that have a "closed form" expression, where there are a finite set of terms. I want to express this too but I don't know how to do that.

Morgan Rogers (he/him) (Jun 27 2021 at 10:17):

I think Liouville proved some such results for complex functions. @Joshua Wrigley was it you who mentioned working on this?

Joshua Wrigley (Jun 27 2021 at 11:29):

Liouville's theorem gives necessary and sufficient conditions for an elementary function (read closed form) to have an elementary integral. The most natural way of understanding elementary functions in that context is in terms of field extensions. Start off with $\mathcal{C}[x]$, polynomials in $\mathbb{C}$, which naturally comes attached with a differential operator $D \colon \mathbb{C} \to \mathbb{C}, \, f(x) \mapsto f'(x)$. We can then add transcendental extensions $\mathbb{C}[x][t]$ where we take $D(t) = f'(x)\cdot t$, and thus $t = A e^{f(x)}$, and similarly for logarithms. But we can repeat the construction again to get expressions like $e^{e^{f(x)}}$. Obviously you can do the same starting from an abstract field with a differential operator. Here's a paper on the subject that's quite readable.

Jacques Carette (Jun 27 2021 at 13:05):

@Ben Sprott it is easy to see that "most" functions in $\mathbb{R}\rightarrow\mathbb{R}$ do not have closed-forms, for any reasonable definition of closed form (i.e. any kind of avatar for the intuitive "can be written down"), purely for cardinality reasons.

Reasonable definitions of "closed form" then start from a basic vocabulary of both base functions and operators to make new ones. One issue that arises very quickly is that many choices lead to undefinedness. And, of course, equality becomes undecidable very quickly! Although, with clever choices (see the work on holonomic functions by van der Hoeven), decidability exists for an extremely large set of functions used in practice.

There really is a vast literature on this topic. For the fascinating sub-problem of expressibility of integrals (or solutions of ODEs), there are a lot of papers. They depend crucially on your choice of basic vocabulary - weird choices lead to very inelegant 'structure theorems', unlike the nice ones of Liouville.

I somehow have serious doubts that category theory will bring new insights to this problem.

John Baez (Jun 27 2021 at 19:06):

For starters category theory is likely to clarify the universal properties of various categories where the objects are the sets $\mathbb{R}^n$ and the morphisms are "functions in closed form", for various definitions of "closed form".

John Baez (Jun 27 2021 at 19:08):

For example I'd expect any reasonable category of this sort to be cartesian, with $\mathbb{R}^n \times \mathbb{R}^m = \mathbb{R}^{n+m}$. But that means it's a Lawvere theory! So, we are looking at various Lawvere theories here.

John Baez (Jun 27 2021 at 19:10):

A very simple, limited concept of "closed form" is "polynomial". This gives a Lawvere theory where the generating operations are the ring operations together with elements of $\mathbb{R}$, which are unary operations. (We can either use all elements, or whatever elements you deem to be in "closed form".)

John Baez (Jun 27 2021 at 19:11):

The story goes on from there. For a while, one would simply be taking results from the vast literature @Jacques Carette is alluding to and giving them a nice categorical packaging. But one might come across some interesting questions this way!