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I have a theorem which requires a lot of notation and explanation and hypotheses in order to state. I usually tend to put all theorem statements in the introduction of the paper so that someone can skim through the introduction to know what the paper is about. I am not sure how to put this very long theorem in the introduction without making the introduction unreadable.
Is there a standard procedure to follow in such cases?
Generally, I feel that the common approach is to sweep the details under the rug, and present a high level version of the theorem. For example:
And finally, we prove that under certain assumptions about the automorphism group of a graph, G, that either G is “essentially” a Johnson graph or it has what we call the Splitting Property. This allows us to construct a new proof of Babai’s proof that GI is in quasipolynomial time that doesn’t rely on the classification theorem of finite groups.
What is meant by “”essentially” as Johnson graph,” or what the “Splitting Property” is will be defined later, but they need not be explained in the introduction. The important thing is to give a gloss explains why someone would want to keep reading. For another example,
We then define a particular distance metric on circle and prove that, under certain modality assumptions, that this distance metric captures the intuitive concept of roundness.
You can also provide pointers to later in the paper where words like “a particular distance metric” or “the Splitting property” is defined.
It is very helpful to have all the assumptions of the main theorem written there explicitly, but is very explicit to refer to another place in the text.
Supposing A, B and C are Euclidean domains satisfying the conditions in assumption 3, then any topology that includes them is plain weird (see definition 2).
Here, assumption 3 is essentially a definition; mathematically, it would be the same to define a new concept which applies to triples (A, B, C).
If appropriate, you can use a qualifier like "technical/regularity/topological assumptions", which could make the theorem itself more informative. This is especially relevant if there are several distinct sets of assumptions you need your object to satisfy.
Another approach would be to not give the entire theorem in the introduction, but rather give an interesting corollary or an elegant example, if any such exist, and refer the reader to the full theorem later in the paper.
