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Polynomial parametrization of quadratic Diophantine equations: decision and construction

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Bogdan Grechuk ↗, Ashleigh Ratcliffe ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We give an algorithm that, for any polynomial with integer coefficients of degree at most two, decides whether its integer zero set is a finite union of polynomial families with integer coefficients and constructs such families whenever they exist. The main step is a constructive proof that, for every nondegenerate integral quadratic form QQ isotropic over Q\mathbb{Q} and every integer mm, the integer solutions of Q(x)=mQ(\mathbf{x})=m form a finite union of polynomial families. This proves the nondegenerate case of a result announced without proof by Vaserstein. For fixed QQ, the number of parameters in each family is bounded independently of mm. The new descent argument treats forms in at least five variables with rational Witt index one. We also settle the previously open cases zt−x2−y2=1zt-x^2-y^2=1 and zt−x2−y2=−1zt-x^2-y^2=-1 by combining Larsen--Nguyen's group parametrization with a classical reduction of binary Hermitian forms, obtaining two and three polynomial families, respectively.

Provenance statement

The proofs were developed through an extended collaboration between the authors and ChatGPT. After partial results obtained by the authors and earlier models, the final step in the proof was obtained with ChatGPT~6~Astra. The main text was drafted with ChatGPT~6~Astra under the authors' guidance; the introduction was written mainly by the authors and received language polishing from AI. Referee-style reports generated by Claude informed the final revisions.

Tools used

OpenAI
ChatGPT
Anthropic
Claude Opus

References

  1. B. Berggren, Pytagoreiska trianglar , Tidskrift för elementär matematik, fysik och kemi 17 (1934), 129--139.
  2. A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups , Annals of Mathematics (2) 75 (1962), 485--535. doi: 10.2307/1970210 (https://doi.org/10.2307/1970210).DOI
  3. D. Carter and G. Keller, Bounded elementary generation of $\SL_n(\OO)$ , American Journal of Mathematics 105 (1983), no. 3, 673--687. doi: 10.2307/2374319 (https://doi.org/10.2307/2374319).DOI
  4. D. A. Cox, Primes of the Form x2+ny2x^2+ny^2: Fermat, Class Field Theory, and Complex Multiplication , second edition, Pure and Applied Mathematics, Wiley, Hoboken, NJ, 2013. doi: 10.1002/9781118400722 (https://doi.org/10.1002/9781118400722).DOI
  5. M. Eichler, Quadratische Formen und orthogonale Gruppen , Grundlehren der mathematischen Wissenschaften 63, Springer-Verlag, Berlin--Göttingen--Heidelberg, 1952. doi: 10.1007/978-3-662-01212-3 (https://doi.org/10.1007/978-3-662-01212-3).DOI
  6. J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory , Springer Monographs in Mathematics, Springer, Berlin, 1998. doi: 10.1007/978-3-662-03626-6 (https://doi.org/10.1007/978-3-662-03626-6).DOI
  7. I. V. Erovenko and A. S. Rapinchuk, Bounded generation of SS-arithmetic subgroups of isotropic orthogonal groups over number fields , Journal of Number Theory 119 (2006), no. 1, 28--48. https://arxiv.org/abs/math/0508480.link
  8. S. Frisch, Remarks on polynomial parametrization of sets of integer points , Communications in Algebra 36 (2008), no. 3, 1110--1114. doi: 10.1080/00927870701776938 (https://doi.org/10.1080/00927870701776938).DOI
  9. S. Frisch and G. Lettl, Polynomial parametrization of the solutions of Diophantine equations of genus 0 , Functiones et Approximatio Commentarii Mathematici 39 (2008), no. 2, 205--209. doi: 10.7169/facm/1229696571 (https://doi.org/10.7169/facm/1229696571).DOI
  10. S. Frisch and L. Vaserstein, Parametrization of Pythagorean triples by a single triple of polynomials , Journal of Pure and Applied Algebra 212 (2008), no. 1, 271--274. doi: 10.1016/j.jpaa.2007.05.019 (https://doi.org/10.1016/j.jpaa.2007.05.019).DOI
Show all 37 referencesShow fewer references
  1. S. Frisch and L. Vaserstein, Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples , Journal of Pure and Applied Algebra 216 (2012), no. 1, 184--191. doi: 10.1016/j.jpaa.2011.06.002 (https://doi.org/10.1016/j.jpaa.2011.06.002).DOI
  2. L. H. Gallardo and L. N. Vaserstein, Polynomial parametrization of integer matrices with given determinant , Journal of Pure and Applied Algebra 218 (2014), no. 2, 321--322. doi: 10.1016/j.jpaa.2013.06.003 (https://doi.org/10.1016/j.jpaa.2013.06.003).DOI
  3. B. Grechuk, On the smallest open Diophantine equations , ACM SIGACT News 53 (2022), no. 1, 36--57. doi: 10.1145/3532737.3532743 (https://doi.org/10.1145/3532737.3532743). Author's version: https://arxiv.org/abs/2204.11624v1 (2022).DOI
  4. B. Grechuk, A systematic approach to Diophantine equations: open problems , arXiv preprint, arXiv:2404.08518v4, 23 January 2026. https://arxiv.org/abs/2404.08518v4.arXiv
  5. B. Grechuk, Polynomial Diophantine Equations: A Systematic Approach , Springer, Cham, 2024. doi: 10.1007/978-3-031-62949-5 (https://doi.org/10.1007/978-3-031-62949-5).DOI
  6. V. Gritsenko, K. Hulek and G. K. Sankaran, Abelianisation of orthogonal groups and the fundamental group of modular varieties , Journal of Algebra 322 (2009), no. 2, 463--478. doi: 10.1016/j.jalgebra.2009.01.037 (https://doi.org/10.1016/j.jalgebra.2009.01.037).DOI
  7. F. J. Grunewald and D. Segal, The solubility of certain decision problems in arithmetic and algebra , Bulletin of the American Mathematical Society (N.S.) 1 (1979), no. 6, 915--918. doi: 10.1090/S0273-0979-1979-14692-5 (https://doi.org/10.1090/S0273-0979-1979-14692-5).DOI
  8. F. Grunewald and D. Segal, Some general algorithms. I: Arithmetic groups , Annals of Mathematics (2) 112 (1980), no. 3, 531--583. doi: 10.2307/1971091 (https://doi.org/10.2307/1971091).DOI
  9. F. J. Grunewald and D. Segal, How to solve a quadratic equation in integers , Mathematical Proceedings of the Cambridge Philosophical Society 89 (1981), no. 1, 1--5. doi: 10.1017/S030500410005787X (https://doi.org/10.1017/S030500410005787X).DOI
  10. R. K. Guy, Conference problems session conducted by J. L. Selfridge , in Number Theory (Ottawa, 1996), R. Gupta and K. S. Williams (eds.), CRM Proceedings & Lecture Notes 19 , American Mathematical Society, Providence, RI, 1999, 385--390; Beukers's Problem CNTA 5.15, p. 390.
  11. B. W. Jordan and Y. Zaytman, On the bounded generation of arithmetic $\SL_2$ , Proceedings of the National Academy of Sciences of the USA 116 (2019), no. 38, 18880--18882. doi: 10.1073/pnas.1907728116 (https://doi.org/10.1073/pnas.1907728116). Correction, 118 (2021), no. 32, e2111531118. doi: 10.1073/pnas.2111531118 (https://doi.org/10.1073/pnas.2111531118). Corrected preprint: https://arxiv.org/abs/1810.12972v3.DOI
  12. M.-A. Knus, Quadratic Forms, Clifford Algebras and Spinors , Seminars in Mathematics 1, IMECC--UNICAMP, Campinas, 1988. https://people.math.ethz.ch/~knus/papers/campinas.pdf.link
  13. M. Larsen and D. Q. N. Nguyen, Polynomial parametrization for $\SL_2$ over quadratic number rings , International Mathematics Research Notices 2021 , no. 9, 6993--7003. doi: 10.1093/imrn/rnz046 (https://doi.org/10.1093/imrn/rnz046). In https://arxiv.org/abs/1808.05349, statements numbered 1.j1.j in the published article are numbered jj.DOI
  14. H. W. Lenstra, Jr., Solving the Pell equation , Notices of the American Mathematical Society 49 (2002), no. 2, 182--192. https://www.ams.org/notices/200202/fea-lenstra.pdf.link
  15. R. Lidl and H. Niederreiter, Finite Fields , 2nd ed., Encyclopedia of Mathematics and its Applications 20, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511525926 (https://doi.org/10.1017/CBO9780511525926).DOI
  16. J. Liu and P. Sarnak, Integral points on quadrics in three variables whose coordinates have few prime factors , Israel Journal of Mathematics 178 (2010), 393--426. doi: 10.1007/s11856-010-0069-y (https://doi.org/10.1007/s11856-010-0069-y).DOI
  17. A. V. Morgan, A. S. Rapinchuk and B. Sury, Bounded generation of $\SL_2$ over rings of SS-integers with infinitely many units , Algebra & Number Theory 12 (2018), no. 8, 1949--1974. doi: 10.2140/ant.2018.12.1949 (https://doi.org/10.2140/ant.2018.12.1949).DOI
  18. J. Neukirch, Algebraic Number Theory , Grundlehren der mathematischen Wissenschaften 322, Springer, Berlin, 1999. doi: 10.1007/978-3-662-03983-0 (https://doi.org/10.1007/978-3-662-03983-0).DOI
  19. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory , translated by R. Rowen, Pure and Applied Mathematics 139, Academic Press, Boston, 1994.
  20. J.-P. Serre, A Course in Arithmetic , Graduate Texts in Mathematics 7, Springer, New York, 1973. doi: 10.1007/978-1-4684-9884-4 (https://doi.org/10.1007/978-1-4684-9884-4).DOI
  21. V. Shoup, A Computational Introduction to Number Theory and Algebra , second edition, Cambridge University Press, Cambridge, 2008. Author's version 2.5: https://www.shoup.net/ntb/ntb-v2_5.pdf.link
  22. C. L. Siegel, Zur Theorie der quadratischen Formen , Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse II (1972), no. 3, 21--46.
  23. Th. Skolem, Diophantische Gleichungen , Ergebnisse der Mathematik und ihrer Grenzgebiete 5, no. 4, Springer, Berlin, 1938.
  24. L. N. Vaserstein, Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups , Annals of Mathematics (2) 171 (2010), no. 2, 979--1009. doi: 10.4007/annals.2010.171.979 (https://doi.org/10.4007/annals.2010.171.979).DOI
  25. J. Voight, Quaternion Algebras , Graduate Texts in Mathematics 288, Springer, Cham, 2021. doi: 10.1007/978-3-030-56694-4 (https://doi.org/10.1007/978-3-030-56694-4).DOI
  26. H. Weber, Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist , Mathematische Annalen 20 (1882), 301--329. doi: 10.1007/BF01443599 (https://doi.org/10.1007/BF01443599).DOI
  27. U. Zannier, Remarks on a question of Skolem about the integer solutions of x1x2−x3x4=1x_1x_2-x_3x_4=1 , Acta Arithmetica 78 (1996), no. 2, 153--164. doi: 10.4064/aa-78-2-153-164 (https://doi.org/10.4064/aa-78-2-153-164).DOI

Version history

  1. v1Submitted by Bogdan GrechukInitial depositCurrentOct 29, 2026