# What to do with long equation-heavy solutions?

I am writing a paper about a problem that involves solving a certain equation. The problem is that the solution is very long and there are a lot of terms. My question is: Is there a way to present the solution in an attractive manner instead of just typing the solution, which is about a page long? Also, would the fact that the solution is long and looks unattractive affect whether the paper get accepted or not?

• This is really a mathematics question. Often, this issue can be improved by introducing good notation. – David Ketcheson Mar 21 '16 at 8:46
• @DavidKetcheson, I tried but the closest I got was about a page long solution. – MrDi Mar 21 '16 at 8:47
• One page doesn't sound like a lot to me. As a rule of thumb, the solution should be as readable as possible. If a reviewer has a simple idea to improve it substantially, he/she is more likely to recommend rejection. In many cases, improving readability involves splitting the solution into parts (e.g., individual lemmas) such that the proofs are relatively short. Then, you add a story that explains the connection of the Lemmas and end with a summary corollary/proposition/lemma/theorem/whatever. – DCTLib Mar 21 '16 at 9:49
• To those voting to close: this is not a dupe of "How best to present long equations in two-column papers?", that question is talking about formatting a single, long equation when you have little horizontal space. This is talking more about presentation of a complex solution. – eykanal Mar 21 '16 at 14:35
• While you may not find long formulae attractive, some readers may at least find them impressive. – Kimball Mar 21 '16 at 20:24

This happens not just in mathematics but in many other disciplines as well: modeling of biological systems, for example, often involves a vast number of terms in the equations. The best approach that I have found for these sorts of situations is to look for the structure in the system and to build your presentation around that structure.

In most cases that I have encountered, it is not true that there are simply a bazillion essentially unrelated terms. The true story of the equation, then, is not the expanded mathematical form but the process and relations that generate it. This opens up a number of approaches for factoring out the equations in order to make them more tractable to present, including:

• Abstracting sub-structures as variables (e.g., X^2+XY+Y^2, where X and Y are complex terms presented separately)
• Separating variables from parameters (e.g., a reaction network where each reaction uses one of several standard Hill equation models, with the parameters of the models presented in a table)

These sorts of factorings also lead to a much more informative and interesting presentation of the equation as well. In fact, the Big Equation itself may often end up being relegated to an appendix, where its page-busting form is of little concern.

There may be, of course, certain systems in which such factoring is impossible---and that fact is quite interesting and worth clear discussion!

• I advocate the idea of putting the expanded equation in an appendix. The meaning, or the story of the equation is what we need most of the time. – Ooker Mar 22 '16 at 6:08

Quite often, following @jakebeal, a formula with many terms can be factored, or split into groups with similar interpretation: terms with similar powers, or groupings that are consistent (for instance in convective and diffusive terms with differential equations). You may put the emphasis on the most important terms, and group the negligible ones. If you are a LaTeX user, you may use different font sizes: How can I change the font size in math equations? to help the reader focus on important parts.

Or if your equation is parametric, and your paper only requires a subset of solutions, those can be given in the text, the rest can be, as suggested, put aside in an appendix.

If you want montruous examples, you can have a look at What is the longest equation known? 