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Disproof of minimum CAD existence for algebraic sets in dimensions n≥4n\ge4

A mix of human-written and AI-generated textHuman understanding: some partscs.SC — Symbolic Computationmath.AG — Algebraic Geometry

Contributed by Qiyue Tang ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We give real algebraic counterexamples to the existence of a minimum adapted cylindrical algebraic decomposition in every dimension n≥4n\ge4, for a fixed coordinate order.

Provenance statement

The main results and proofs were developed using GPT-6-Pro. The archived research session lasted 1 hour, 55 minutes, 29 seconds, measured from research start to packaging. Codex was used to audit the arguments and polish the manuscript. The manuscript passed an AI review using Rethlas-OSV.

References

  1. Lucas Michel, Pierre Mathonet, and Naïm Zénaïdi, On Minimal and Minimum Cylindrical Algebraic Decompositions , ISSAC 2024; arXiv:2411.13218. https://arxiv.org/abs/2411.13218.arXiv
  2. Lucas Michel, Pierre Mathonet, and Naïm Zénaïdi, Further results on Minimal and Minimum Cylindrical Algebraic Decompositions , arXiv:2601.09548, 2026; Journal of Symbolic Computation, DOI 10.1016/j.jsc.2026.102587. https://arxiv.org/abs/2601.09548.arXivDOI
  3. Lucas Michel, On Minimum CADs for Algebraic Sets in Dimension Three , ISSAC 2026, pp. 281--289, DOI 10.1145/3815436.3815460; arXiv:2605.04718. https://arxiv.org/abs/2605.04718.arXivDOI
  4. Daniel Pecker, On the elimination of algebraic inequalities , Pacific Journal of Mathematics 146 (2), 1990, pp. 305--314. https://msp.org/pjm/1990/146-2/pjm-v146-n2-p07-s.pdf.link
  5. OSVerify contributors, Rethlas-OSV , software repository, 2026. https://github.com/OSVerify/Rethlas-OSV.link
  6. Fan Nie, Haotian Ye, Shurui Liu, Zihao Wang, Stefano Ermon, and James Zou, One-Sided Verification Under Limited Expert Review , arXiv:2610.18089v2, 2026. https://arxiv.org/abs/2610.18089v2.arXiv

Version history

  1. v1 · 18 pagesSubmitted by Qiyue TangInitial depositCurrentNov 05, 2026