This advice is relevant to articles written specifically for mathemtics or statistics journals. For guidance relevant to publications that use mathematics but are not part of the mathematics literature, see Journal articles.

Each journal has instructions for authors (sometimes titled ‘author guidelines’ or ‘advice to authors’). The instructions cover many different areas, from formatting and maximum word limits to copyright issues and procedures for submitting papers. Mathematics journals often provide a LaTeX template for authors to use – use it!

Structuring a mathematics paper

Mathematics papers are somewhat variable in structure, but often use some subset of these sections:

• Introduction. What was the problem or question? Why study it?
• Preliminary considerations and definitions. What are the key symbols and terminologies used? What are the main assumptions?
• Results and discussion. What did we find? This traces the evolution of the study, and may consist of several sections, each with a descriptive heading (eg Properties of the second-order series).
• Main results/conclusions/future work. What key results or theorems have we shown? What are the implications? Where to next?

This structure helps you, as the author, to organise your information; it will also help the reader to obtain information efficiently.

The introduction should pair with the concluding material. In other words, the conclusion should respond to the questions asked in the introduction.

Other components of a journal paper may include:

• abstract
• author names and affiliations
• keywords
• abbreviations
• acknowledgments
• conflict-of-interest statement
• appendixes
• references
• additional online material (eg code, other results).

Writing a mathematics paper

Mathematics papers are in a sense best written backwards.

Step 1: Determine the purpose and the content of the paper

Start with the final result. This could be a proof or disproof of a conjecture, a new method for tackling a problem, an improved algorithm, and so on. What do you want your readers to remember?

This guides further content in 2 ways:

• What must the reader be told if they are to appreciate the broader significance of the result? Provide a paragraph or two of context in the introduction. Consider writing it in a way a nonspecialist can understand. This promotes cross-disciplinary links.
• How can the chain of reasoning that leads to the result be most clearly presented? What steps in any derivations are essential and must be shown? Can diagrams assist the reader? Is there a choice of notation? (Is the notation you used in your research work the best one to use when publishing?) Is there useful supplementary material, such as computer code or 3D visualisations of surfaces, that you might want to prepare and deposit?

Step 2: Write the body of the paper

Once  these questions are answered, you are ready to assemble the material, order it, and link the steps together with clear language. In some publications, this means a formal string of numbered propositions, lemmas, theorems and proofs. Others are more flexible, but a few key points are as follows:

• Be clear. Use unambiguous language and choose unambiguous symbols. Do not drop subscripts just to simplify writing the paper. Be clear and explicit at all times.
• Be logical. Every step must flow logically from the one before, and the reader must be able to follow this themselves. Although the mathematics must be complete, do not be afraid to use words and diagrams as well as mathematics to explain how to get from A to B.
• Use definitions. The reader must never be in doubt as to what a symbol means, what axioms have been assumed and what unfamiliar terminology refers to. Formal definitions, numbered and with headings, help a reader when they are unsure.
• Adhere to formatting conventions. Theorems, propositions and lemmas in journals are often numbered and and given subheadings, perhaps italicised, to clearly separate them from the discussions and derivations. Doing this gives a paper the correct ‘look’, and will help experienced readers find what they seek. Use bold, italic and other fonts as prescribed. Check carefully for existing conventions before introducing new symbols or notation. Cite and reference according to the established style.

If in doubt, review some issues of the journal you will be submitting to.

Step 3: Polish your final draft

Review the text for language, correctness and clarity. Check that the references are complete and correctly presented.

Assemble the front matter: author details, title, abstract, keywords, possibly discipline area codes, and so on.

Assemble the back matter: acknowledgments (including listing any relevant grant numbers and funding sources), additional material, appendixes.

Of these, the most important are the abstract and title:

• Abstract. The abstract must be easy to read and understand. It is the first thing that a journal editor and reviewer will read. If the paper is published, the abstract will be the part that readers use to decide whether to read the rest of the paper. Summarise each of the main sections of the paper in 1 or more sentences, so write it after the paper is complete. When writing the abstract
  • be brief, follow the usual structure for the specific journal, and ensure that the abstract is easy to read and understand on its own
  • do not include citations, references to tables and figures, or abbreviations (unless a term is used several times).
• Title. Although it is good to have a working title, you can leave the final version until you have written the rest of the paper. The title should be informative and fit with any length restrictions.