I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
/

Exact Quotients of Fresnel–Kummer Surfaces and Certified Biaxial Refraction

Tanush Shaska

I-AIMS preprint sh-180 · 2026

The Fresnel wave surface governs the propagation of light in a transparent biaxial crystal. It is a special Kummer quartic, and we identify it exactly. Over the complex numbers the wave surface of a crystal is the Kummer surface of the Jacobian of an explicit genus-two curve branched at the signed square roots of the three principal permittivities. This Jacobian is isogenous, by an isogeny with kernel of order four, to a product of two elliptic curves. One elliptic curve carries the three permittivities, and the other carries the optic-axis angle. The physical family is Zariski dense in the locus of genus-two curves with an extra involution, and its automorphism strata are explicit. The identification instantiates a task-aware quotient, which identifies parameters that differ by a nuisance transformation and carries invariant coordinates and explicit strata. For biaxial crystals, two ratios of the permittivities form a complete invariant of the wave surface up to rotation and rescaling, and the four real nodes are given in closed form. At an interface the candidate transmitted waves are the roots of a quartic of exact degree four. Its real-root count, root order, and repeated-root events are decided by exact algebraic predicates, and along the generic single-node encounters of the paper its discriminant vanishes to second order. Floating-point solvers drop forward transmitted modes near the optic axes, and the certified solver does not. On exact equivalence classes, learned models on quotient coordinates are invariant and more accurate than models on raw tensors, while learned root-count predicates fail near the optic axes.

MSC classes: Primary 14H45; Secondary 14J28, 14Q10, 78A05, 65H04, 68T07

References

  1. P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, The Kummer tensor density in electrodynamics and in gravity, Annals of Physics, 349 (2014), pp. 297–324.
  2. C. Birkenhake and H. Lange, Complex Abelian Varieties, vol. 302 of Grundlehren der mathematischen Wissenschaften, Springer, second ed., 2004.
  3. B. Blum-Smith and S. Villar, Machine learning and invariant theory, Notices of the American Mathematical Society, 70 (2023), pp. 1205–1213.
  4. O. Bolza, On binary sextics with linear transformations into themselves, American Journal of Mathematics, 10 (1887), pp. 47–70.
  5. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, seventh ed., 1999.
  6. S. P. Bos, S. Y. Haffert, and C. U. Keller, Rigorous vector wave propagation for arbitrary flat media, in Polarization Science and Remote Sensing VIII, vol. 10407 of Proc. SPIE, 2017, p. 1040709, https://doi.org/10.1117/12.2273341. arXiv:1811.09777.
  7. M. Boutin and P.-L. Bazin, Structure from motion: a new look from the point of view of invariant theory, SIAM Journal on Applied Mathematics, 64 (2004), pp. 1156–1174.
  8. M. M. Bronstein, J. Bruna, T. Cohen, and P. Veličković, Geometric deep learning: grids, groups, graphs, geodesics, and gauges, 2021. arXiv:2104.13478.
  9. N. Bruin and K. Doerksen, The arithmetic of genus two curves with (4,4)-split Jacobians, Canadian Journal of Mathematics, 63 (2011), pp. 992–1024.
  10. K. G. Budden, The Propagation of Radio Waves, Cambridge University Press, 1985.
  11. J. W. S. Cassels and E. V. Flynn, Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, vol. 230 of London Mathematical Society Lecture Note Series, Cambridge University Press, 1996.
  12. A. Clingher, A. Malmendier, and T. Shaska, Kummer shadows of elliptic subcovers, 2026. preprint.
  13. A. Cocan, M. V. de Hoop, J. Ilmavirta, M. Lassas, and A. Várilly-Alvarado, Generic recovery of permittivity and permeability in anisotropic Maxwell systems, 2026. arXiv:2608.02900.
  14. G. E. Collins and A. G. Akritas, Polynomial real root isolation using Descartes’ rule of signs, in Proceedings of the Third ACM Symposium on Symbolic and Algebraic Computation (SYMSAC ’76), 1976, pp. 272–275.
  15. I. Dolgachev, Lectures on Invariant Theory, Cambridge University Press, 2003.
  16. A. Favaro and F. W. Hehl, Fresnel versus Kummer surfaces: geometrical optics in dispersionless linear (meta)materials and vacuum, 2014. arXiv:1401.4077.
  17. A. Favaro and F. W. Hehl, Light propagation in local and linear media: Fresnel–Kummer wave surfaces with 16 singular points, Physical Review A, 93 (2016), p. 013844.
  18. R. W. H. T. Hudson, Kummer’s Quartic Surface, Cambridge University Press, 1905.
  19. G. Humbert, Sur les fonctions abéliennes singulières. I, II, III, Journal de Mathématiques Pures et Appliquées (5), 5 (1899), 233–350; 6 (1900), 279–386; 7 (1901), 97–123 (1899–1901).
  20. J.-i. Igusa, Arithmetic variety of moduli for genus two, Annals of Mathematics, 72 (1960), pp. 612–649.
  21. E. Kani, Elliptic curves on abelian surfaces, Manuscripta Mathematica, 84 (1994), pp. 199–223.
  22. E. Kani, The number of curves of genus two with elliptic differentials, Journal für die reine und angewandte Mathematik, 485 (1997), pp. 93–122.
  23. A. Kumar, Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields, Research in the Mathematical Sciences, 2 (2015). Article 24, 46 pp.
  24. P. Latorre, F. J. Seron, and D. Gutierrez, Birefringence: calculation of refracted ray paths in biaxial crystals, The Visual Computer, 28 (2012), pp. 341–356.
  25. K. Magaard, T. Shaska, and H. Völklein, Genus 2 curves that admit a degree 5 map to an elliptic curve, Forum Mathematicum, 21 (2009), pp. 547–566.
  26. R. R. McLeod and K. H. Wagner, Vector Fourier optics of anisotropic materials, Advances in Optics and Photonics, 6 (2014), pp. 368–412.
  27. L. Mescheder, M. Oechsle, M. Niemeyer, S. Nowozin, and A. Geiger, Occupancy networks: learning 3D reconstruction in function space, in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 4460–4470.
  28. J.-F. Mestre, Construction de courbes de genre 2 à partir de leurs modules, in Effective Methods in Algebraic Geometry, vol. 94 of Progress in Mathematics, Birkhäuser, 1991, pp. 313–334.
  29. D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Springer, third ed., 1994.
  30. V. V. Nikulin, On Kummer surfaces, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 39 (1975), pp. 278–293. English translation: Math. USSR-Izv. 9 (1975), 261–275.
  31. M. Ovsjanikov, Q. Mérigot, V. Pătrăucean, and L. Guibas, Shape matching via quotient spaces, Computer Graphics Forum, 32 (2013).
  32. J. J. Park, P. Florence, J. Straub, R. Newcombe, and S. Lovegrove, DeepSDF: learning continuous signed distance functions for shape representation, in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 165–174.
  33. E. L. Rees, Graphical discussion of the roots of a quartic equation, The American Mathematical Monthly, 29 (1922), pp. 51–55.
  34. F. Rouillier and P. Zimmermann, Efficient isolation of polynomial’s real roots, Journal of Computational and Applied Mathematics, 162 (2004), pp. 33–50.
  35. D. E. Rowe, Models from the nineteenth century used for visualizing optical phenomena and line geometry, in Model and Mathematics: From the 19th to the 21st Century, M. Friedman and K. Krauthausen, eds., Trends in the History of Science, Birkhäuser, 2022. arXiv:1912.07138.
  36. A. Schulz, J. Xu, B. Zhu, C. Zheng, E. Grinspun, and W. Matusik, Interactive design space exploration and optimization for CAD models, ACM Transactions on Graphics, 36 (2017).
  37. E. Shaska and T. Shaska, Machine learning for moduli space of genus two curves and an application to isogeny-based cryptography, Journal of Algebraic Combinatorics, 61 (2025), p. Paper No. 23, https://doi.org/10.1007/s10801-025-01393-8. arXiv:2403.17250.
  38. T. Shaska, Curves of genus 2 with (n, n) decomposable Jacobians, Journal of Symbolic Computation, 31 (2001), pp. 603–617.
  39. T. Shaska and H. Völklein, Elliptic subfields and automorphisms of genus 2 function fields, in Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), Springer, 2004, pp. 703–723.
  40. S. Steinberg, Analytic spectral integration of birefringence-induced iridescence, Computer Graphics Forum, 38 (2019).
  41. V. Vavryčuk, Ray tracing in anisotropic media with singularities, Geophysical Journal International, 145 (2001), pp. 265–276.
  42. S. Villar, D. W. Hogg, K. Storey-Fisher, W. Yao, and B. Blum-Smith, Scalars are universal: equivariant machine learning, structured like classical physics, in Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 28848–28863.
  43. A. Weidlich and A. Wilkie, Realistic rendering of birefringency in uniaxial crystals, ACM Transactions on Graphics, 27 (2008), pp. 1–12.
  44. Y.-L. Yang, Y.-J. Yang, H. Pottmann, and N. J. Mitra, Shape space exploration of constrained meshes, ACM Transactions on Graphics, 30 (2011).
  45. M. Yavartanoo, J. Chung, R. Neshatavar, and K. M. Lee, 3DIAS: 3D shape reconstruction with implicit algebraic surfaces, in IEEE/CVF International Conference on Computer Vision (ICCV), 2021, pp. 12446–12455.
  46. W.-Q. Zhang, General ray-tracing formulas for crystal, Applied Optics, 31 (1992), pp. 7328–7331.

All preprints