How to Write Your First Mathematics Research Paper

The Milestone

Writing your first mathematics research paper is a milestone. It marks the transition from learning existing mathematics to contributing new mathematics to the body of human knowledge.

This guide will walk you through the process, from the moment you have a result worth writing up to the moment you submit your paper.

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What Counts as a Research Paper?

A mathematics research paper presents new results — theorems, proofs, constructions, counterexamples, or algorithms — that advance mathematical knowledge. It differs from an expository article or a thesis chapter in that it must contain something genuinely new.

However, "new" does not mean "earth-shattering." A publishable contribution might be:

• A new proof of a known result that is simpler, shorter, or more illuminating
• A generalization of an existing theorem to a broader setting
• A counterexample to a conjecture or to a natural-sounding claim
• A connection between two previously unrelated areas
• A new construction or algorithm
• A computational verification or extension of existing results

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Before You Start Writing

Verify Your Results

Before writing a single word, make absolutely sure your results are correct.

• Check every proof line by line.
• Look for gaps — places where you claim something follows but have not verified the details.
• Test your theorems against known examples and edge cases.
• If possible, discuss your results with your advisor, a collaborator, or a knowledgeable colleague.

Submitting a paper with an error is far worse than delaying submission to double-check.

Know the Literature

Before writing, you must know what has already been done. Search the relevant literature thoroughly:

• Use MathSciNet or zbMATH for comprehensive literature searches.
• Search arXiv for recent preprints that may not yet be in databases.
• Read the references in closely related papers — they often lead to earlier work you may have missed.

You need to be able to state clearly: "Here is what was known before. Here is what this paper adds."

Identify Your Audience

Who will read this paper? Specialists in your subfield? A broader mathematical audience? This affects your level of exposition, the amount of background you include, and the notation you choose.

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The Structure of a Mathematics Paper

Most mathematics papers follow a standard structure:

1. Title

Choose a title that is specific and informative. "On certain properties of functions" is too vague. "A sharp bound for the chromatic number of triangle-free graphs" tells the reader exactly what the paper is about.

2. Abstract

The abstract is a brief summary (typically 100–200 words) of the paper's main results. It should state the problem, the main theorem or contribution, and, if space allows, the key technique.

3. Introduction

The introduction does several things:

• Motivates the problem. Why is this question interesting? Where does it come from?
• States the main results. Give precise theorem statements, even if they are repeated later.
• Puts the results in context. How do they relate to existing work?
• Outlines the structure of the paper. "In Section 2, we establish notation and recall..."

The introduction is the most important part of the paper. It is what most readers will read first, and many will read only the introduction.

4. Preliminaries

Define notation and recall the definitions and theorems you will use. Give references for all cited results. This section should be concise — include only what is needed.

5. Main Results

Present your theorems and their proofs. Organize logically: lemmas first, then the main theorem, then corollaries.

Each theorem should be clearly stated before its proof begins. Each proof should begin by outlining its strategy.

6. Examples and Applications (if applicable)

Concrete examples illustrating your results are extremely valuable. They help readers understand the theorems and verify them informally.

7. Concluding Remarks or Open Questions

Discuss what your results do not resolve. Are there natural generalizations? Open conjectures? This section shows that you understand the broader landscape.

8. References

A complete, carefully formatted bibliography. Use BibTeX and follow the conventions of your target journal.

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Writing Principles

Write for the Reader

Paul Halmos gave this advice:

Write as if you are explaining your work to a colleague — clearly, precisely, but without unnecessary formality.

Motivate Everything

Before stating a definition, explain why it is needed. Before stating a theorem, explain what it says informally. Before giving a proof, sketch the idea.

The reader should never wonder "why are we doing this?"

Be Precise

Mathematics demands precision. Define every term. State every hypothesis. Distinguish between "for all" and "there exists." Make sure your quantifiers are in the right order.

Use Standard Notation

Follow the conventions of your field. If $\mathbb{R}^n$ denotes Euclidean space in your area, do not use $R^n$. If homology groups are written $H_n(X)$, do not write $\mathcal{H}_n(X)$ without good reason.

Short Sentences, Short Paragraphs

Mathematical prose should be easy to parse. Long, convoluted sentences are the enemy of clarity.

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Writing Proofs

The Three Parts of a Proof

Every proof has three components:

1. Setup: State what you are proving and establish notation.
2. Argument: The logical chain of deductions.
3. Conclusion: Explicitly state what has been proved.

Common Proof-Writing Mistakes

• Leaving gaps. Every step should follow logically from the previous steps and the stated hypotheses.
• Assuming what you want to prove. This is the most serious error. Be vigilant about circular reasoning.
• Using notation before defining it.
• Claiming "it is easy to see" for non-obvious steps. Either prove it or give a reference.
• Not indicating the proof strategy. A sentence like "We proceed by induction on $n$" or "Suppose for contradiction that..." helps the reader enormously.

Jean-Pierre Serre's advice is worth remembering:

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The Revision Process

First Draft

Get your ideas down on paper. The first draft will be rough, and that is fine. Focus on correctness and completeness rather than elegance.

Second Draft

Reorganize for clarity. Improve the exposition. Add motivation and examples. Cut unnecessary material.

Third Draft and Beyond

Polish the prose. Check every proof again. Ensure consistency of notation. Read the paper aloud — this catches awkward phrasing that your eyes might skip.

Get Feedback

Before submitting, ask colleagues or your advisor to read the paper. Fresh eyes catch errors and unclear passages that you have become blind to.

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Where to Submit

Choosing a Journal

Consider:

• Scope: Does the journal publish papers in your area?
• Quality level: Is your result appropriate for the journal's standards? It is better to aim accurately than to waste time with inappropriate submissions.
• Audience: Will the journal's readers be interested in your work?
• Turnaround time: Some journals take months; others take years. Ask your advisor for guidance.

The arXiv

Before or simultaneously with journal submission, most mathematicians post their papers to arXiv. This makes your work immediately available to the community and establishes priority.

The Review Process

After submission, your paper will be assigned to a referee (or referees) who will evaluate it for correctness, significance, and exposition. This process can take months or even years.

The referee's report will typically:

• Accept the paper as is (rare for a first submission)
• Request revisions (common — this is a good outcome)
• Reject the paper (with comments that may help you improve it or suggest a different journal)

Rejection is not failure — it is a normal part of the process. Most published mathematicians have had papers rejected.

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LaTeX and Formatting

• Use a clean LaTeX template. The amsart document class is standard for mathematics.
• Number all theorems, lemmas, propositions, and equations.
• Use \label and \ref for cross-references.
• Format your bibliography with BibTeX using amsplain or a similar style.
• Follow the journal's style guide if one is provided.

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Common Mistakes in First Papers

1. The introduction is too short. Your introduction should be substantial — often 1–2 pages.
2. The abstract restates the introduction. The abstract should stand alone.
3. Background material is excessive. Include only what the reader needs.
4. The paper lacks examples. Good examples make your results accessible.
5. References are incomplete. Cite all relevant prior work.

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Final Encouragement

Writing your first paper is difficult, and it takes longer than you expect. But it is also one of the most satisfying experiences in a mathematical career.

Every mathematician's first paper was imperfect. What matters is that the mathematics is correct, the exposition is clear, and the contribution is honest.

As the great mathematician Paul Erdős said:

The theorems are the hard part. Writing them up clearly is the final, essential step.

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References

• Paul Halmos, "How to Write Mathematics," L'Enseignement Mathématique, Vol. 16, 1970
• Nicholas Higham, Handbook of Writing for the Mathematical Sciences, SIAM, 1998
• Steven Krantz, A Primer of Mathematical Writing, American Mathematical Society, 2017
• Donald Knuth, Tracy Larrabee, and Paul Roberts, Mathematical Writing, MAA Notes, 1989
• Terence Tao, Advice on writing papers
• arXiv, Submission guidelines
• American Mathematical Society, AMS Author Handbook