Foundations of Computational Mathematics (foundations-of-computational-mathematics)

Journal positioning

Foundations of Computational Mathematics, published by Springer for the Society for the Foundations of Computational Mathematics, is a leading venue for the theory at the interface of mathematics and computation. Its defining character is rigorous mathematical foundations for computation: numerical analysis, approximation and complexity theory, optimization, computational geometry and topology, information-based complexity, and the analysis of algorithms — always with theorems and proofs at the core rather than empirical performance. The journal rewards work that establishes the mathematical basis of computational methods: convergence and error analysis, complexity bounds, stability, and structural theory, addressed to researchers who treat computation as a mathematical subject. It is selective and emphasizes depth and lasting foundational value. This skill is a fit / venue-selection / re-framing tool. It does not replace the journal's current official submission guidelines or the editorial board's judgment. Before submitting, re-check the live author instructions on the FoCM / Springer site.

When to trigger

• The author names FoCM as the target for a result that establishes rigorous mathematical foundations of a computational method or problem.
• A manuscript proves convergence, error, complexity, or stability theory for numerical, optimization, or geometric/topological computation, and the author is choosing between FoCM and SIAM Review or Journal of the ACM.
• A paper is theory-first at the math–computation interface and would be undersold by a venue that prioritizes empirical results.
• The author needs FoCM's expectations on full rigor, exposition, and submission norms before committing.

Scope & topic fit

• Numerical analysis with rigorous theory: convergence, error bounds, and stability of methods for differential equations, linear algebra, and approximation.
• Approximation theory, sampling, and information-based complexity: optimal rates, lower bounds, and the mathematics of recovery.
• Optimization theory: convergence and complexity of continuous and convex/nonconvex optimization, with rigorous guarantees.
• Computational complexity in the real/algebraic and continuous settings, and the analysis of algorithms over continuous structures.
• Computational geometry and topology: persistent homology, topological data analysis, and geometric computation with provable foundations.
• Mathematical foundations of learning, signal recovery, and data analysis where the contribution is theorems and bounds.

Method & evidence bar

• Results must be complete and fully proved: theorems established rigorously with explicit hypotheses, no gaps, and no reliance on empirical evidence in place of proof.
• Foundational significance is required: the contribution must advance the mathematical understanding of a computational method or problem — sharper bounds, new convergence/complexity theory, or a structural insight — not merely a faster implementation.
• Proofs must be referee-verifiable in full; constants, rates, and assumptions must be stated precisely, and lemmas proved or precisely cited.
• Where numerical experiments appear, they illustrate or motivate the theory; they do not substitute for it, and the paper's claims rest on the proofs.
• The Mathematics Subject Classification (MSC) should be assigned accurately; arXiv posting is standard and encouraged, with any computer-assisted proof documented for independent checking. There is no data/code-deposition norm, though making accompanying code available is welcomed.
• Prior and concurrent results must be credited precisely, with the improvement over known rates/bounds stated exactly.

Structure & house style

• Standard mathematical structure: abstract, introduction stating results and significance, preliminaries and setup, the body of theorems and proofs, optional illustrative experiments, and references.
• The introduction must state the main theorems precisely, give the foundational significance, and position the rates/bounds against prior work.
• Exposition must be rigorous and clear: precise definitions, explicit constants and assumptions, and proofs organized so an expert can verify each step.
• Long arguments may be modularized into lemmas and propositions; technical estimates or auxiliary computations may go to appendices.
• Illustrative numerical experiments, if included, must be clearly subordinate to the theory and described reproducibly.
• LaTeX is expected; bibliographic and formatting conventions follow the journal's current style.

Official-submission checklist

• Before giving submission-ready advice, read ../../resources/source-basis.md and ../../resources/official-source-map.md; start from the official source anchors for this journal family, then cite the current journal-specific page you checked.
• Search the live site for "Foundations of Computational Mathematics submission guidelines" and follow the current Springer version.
• Re-check the submission procedure, file-format and LaTeX requirements, and editor-handling conventions.
• Re-check MSC classification requirements and the journal's expectations on exposition, length, and the role of any numerical experiments.
• Re-check authorship, competing-interests, and AI-use disclosure requirements; confirm arXiv/preprint posting policy and documentation expectations for computer-assisted proofs.
• If the live official instructions conflict with this skill, the official instructions win.

Pre-submission self-check

• One sentence — the main theorem and the foundational advance it makes at the math–computation interface.
• Every proof is complete and referee-verifiable, with explicit constants/rates/assumptions and lemmas proved or cited.
• The contribution is theory-first; any numerical experiments illustrate rather than substitute for the proofs.
• The introduction states results precisely and positions the rates/bounds honestly against prior work.
• MSC classification is assigned; arXiv status and any computer-assisted-proof documentation are consistent with policy.
• The result's foundational significance matches FoCM rather than an applications- or empirics-driven venue.

Common desk-reject triggers

• An empirical or applied paper whose contribution is performance rather than proved foundations.
• A result with incomplete proofs, unstated assumptions, or claims resting on experiments instead of theorems.
• An incremental improvement of a known bound without a new method or structural insight of lasting value.
• Overstated significance or imprecise positioning relative to existing rates and complexity results.
• A manuscript whose exposition is too disorganized for a referee to verify the analysis.

Re-routing decision

• A broad, expository, survey-style treatment of a computational-mathematics topic for a wide audience: siam-review.
• A theoretical-computer-science result centered on algorithms and complexity rather than continuous/numerical foundations: journal-of-the-acm or siam-journal-on-computing.
• A pure-mathematics result whose computational framing is incidental: a leading pure-math venue such as acta-mathematica.
• An applied numerical-methods paper driven by application performance: a SIAM applied journal (e.g., SINUM, SISC).

Output format

[Fit] High / Medium / Low (one-line reason)
[Target] Foundations of Computational Mathematics
[Topic tags] <2–3 closest areas / MSC themes>
[Method/evidence] <are the proofs complete and referee-verifiable, and is the contribution a foundational advance at the math–computation interface?>
[Top risk] <the single most likely reason for rejection>
[Official items to re-check] <submission/LaTeX format / MSC classification / role of numerical experiments & length / arXiv & computer-proof documentation / disclosure>
[Re-route suggestion] <if not a fit, a better-matched venue>