The 2010 paper of Perez-Diaz, Sendra, and Villarino gave the first algorithms for deciding whether a rational parametrization of an affine algebraic surface over an algebraically closed field of characteristic zero is normal, that is surjective onto the surface, while explicitly leaving the construction of normal reparametrizations open, together with two robustness gaps of the decision procedures (one-dimensional exceptional sets arising in zero-tests, and the case of nonempty global intersection). Later literature fundamentally reshaped the problem: for plane curves normal parametrizations always exist and are computable, but for surfaces birational surjective parametrizations need not exist (the affine Fermat cubic is a counterexample), and non-existence extends to finite surjective morphisms; the constructive substitute is covering a surface by two or three parametrization images. The surviving open core is precise: characterize exactly which rational affine surfaces admit a normal parametrization (the known infinity-curve and Picard-theoretic conditions are only necessary), give an algorithm that decides existence and computes a normal reparametrization when one exists, and close the robustness gaps of the normality decision procedures themselves.

Background

A rational algebraic surface is a surface birational to the affine plane; a rational parametrization of it is a tuple of rational functions P(s,t)=(p1/q1,p2/q2,p3/q3)\mathcal{P}(s,t)=(p_1/q_1,p_2/q_2,p_3/q_3) whose image is Zariski dense in the surface. The parametrization is proper (birational) if the induced map is generically injective; its tracing index is the degree of the generic fiber, and parametrizations of tracing index one are exactly the proper ones. The parametrization is normal if it is surjective: every point of the surface is the value of P\mathcal{P} at some parameter pair where it is defined. Normality matters wherever a parametric representation is used as if it were the surface itself, for instance in plotting and rendering without missing points, and in algorithms that must reach every point, such as computing the distance from a point to a surface or analyzing unreachable zones of algebraic surfaces attached to differential equations.

For plane curves the theory is complete: every rational affine curve over an algebraically closed field of characteristic zero admits a proper normal parametrization, and Sendra (2002) gave an algorithm reparametrizing any rational parametrization of a curve into a normal one. For surfaces the question is older and harder; Gao and Chou (1991) obtained the first results for special classes such as quadrics. The source paper (Perez-Diaz, Sendra, Villarino, 2010) introduced pseudo-normality, an algebraic necessary condition for normality expressed through the sets Ck=V(pk)\mathcal{C}_k=\mathbb{V}(p_k) and Dk=V(qk)\mathcal{D}_k=\mathbb{V}(q_k) of zeros of numerators and denominators, proved that pseudo-normal polynomial parametrizations are normal, gave an algorithm deciding normality for certain classes of parametrizations, and showed that cylinders over coordinate planes admit normal parametrizations by lifting normal curve parametrizations. It explicitly left open the construction of normal reparametrizations, and flagged two robustness gaps of the decision procedures: one-dimensional exceptional sets that the zero-tests can produce, whose decomposition into irreducible components may be far more complicated than the leading-coefficient locus Z1\mathcal{Z}_1 and may render the algorithm impractical (Remark 3.8); and the case where the global intersection G∗=⋂k(Ck∩Dk)\mathcal{G}^*=\bigcap_k(\mathcal{C}_k\cap\mathcal{D}_k) is nonempty, where for some parameter pairs the induced candidate-point set Z(s0,t0)\mathcal{Z}(s_0,t_0) may be the whole surface (Remark 4.5).

Subsequent work transformed the existence picture. Caravantes, Sendra, Sevilla, and Villarino (2018) proved that if a rational affine complex surface whose projective closure is smooth admits a birational surjective parametrization from an open subset of the affine plane, then the infinity curve (the intersection of the projective closure with the hyperplane at infinity) must contain at least one rational component; consequently the affine Fermat cubic x3+y3+z3=1x^3+y^3+z^3=1, whose infinity curve is a smooth genus-one cubic, admits no birational surjective parametrization, and neither does the hyperboloid, whose rational conic at infinity satisfies the necessary condition but is excluded by a separate morphism-extension argument (Example 4.3 of that paper). Ballico and Fontanari (2022) extended non-existence to finite surjective morphisms f:A2→Sf:\mathbb{A}^2\to S: if a smooth projective rational surface contains an ample curve of positive genus, its complement admits no such morphism, and in general the boundary must contain at least ρ(Sˉ)\rho(\bar S) rational curves generating Pic(Sˉ)⊗Q\mathrm{Pic}(\bar S)\otimes\mathbb{Q}, where ρ\rho is the Picard number, the rank of the group of divisor classes modulo linear equivalence. On the constructive side, the same groups developed covering techniques: every affine rational surface parametrized without projective base points (base points being parameter values, including those at infinity, at which all coordinate functions of the homogenized parametrization vanish simultaneously) is covered by at most three parametrizations (Sendra, Sevilla, Villarino, 2014), refined to two parametrization images in the affine case, optimal in general precisely because single normal parametrizations need not exist (Caravantes et al., 2021), while the projective case needs three charts with optimality left open; every smooth rational projective surface is an open union of three affine planes (Caravantes et al., 2023). In the adjacent properness problem, computing a birational reparametrization of a given unirational surface parametrization (one whose image is dense in the surface but which is not assumed generically injective), a general algorithm now exists (Caravantes, Pérez-Díaz, Sendra, 2024), which contrasts sharply with the still-open surjectivity side.

Problem Statement

Over an algebraically closed field K\mathbb{K} of characteristic zero, given a rational parametrization P(s,t)=(p1/q1,p2/q2,p3/q3)\mathcal{P}(s,t)=(p_1/q_1,p_2/q_2,p_3/q_3) of rational functions in the parameters s,ts,t of an affine algebraic surface S⊂K3\mathcal{S}\subset\mathbb{K}^3, the problem, stated as future work in the source paper, is to make the theory of normal (surjective) parametrizations of surfaces effective. Its common objective is a complete effective correspondence: for every rational surface parametrization, either a certified normal reparametrization or a certified obstruction. Concretely, the objective has three named components, each an independently resolvable sub-goal: resolving any one of them is a substantive advance, while the complete correspondence requires all three. (1) characterize the rational affine surfaces that admit a normal parametrization, meaning a surjective rational parametrization of arbitrary tracing index (the tracing index being the degree of the generic fiber of the parametrization, so that no generically injective behavior is required), through effectively computable invariants; the existing conditions, a rational component in the infinity curve for smooth projective closures in the birational case and the Picard-theoretic boundary conditions for finite surjective morphisms, are known to be necessary only, and the intermediate class of surjective parametrizations that are neither birational nor finite morphisms is not characterized in either direction; (2) given a rational affine surface together with a rational parametrization of it, decide whether a normal parametrization exists and, when it does, compute one by an algorithm with proven termination and correctness, since the 2010 paper decides normality only for a fixed parametrization under hypotheses and explicitly leaves this construction open; (3) close the two robustness gaps of the normality decision procedures themselves, which are separate sub-tasks: the effective treatment of the one-dimensional exceptional sets that the zero-tests in the pseudo-normality and normality algorithms can produce (pseudo-normality being the algebraic necessary condition for normality formulated via the zero sets Ck=V(pk)\mathcal{C}_k=\mathbb{V}(p_k) and Dk=V(qk)\mathcal{D}_k=\mathbb{V}(q_k) of the numerator and denominator of the kk-th coordinate of P\mathcal{P}) (Remark 3.8 of the source), and the case where the global intersection G∗=⋂k(Ck∩Dk)\mathcal{G}^*=\bigcap_k(\mathcal{C}_k\cap\mathcal{D}_k) is nonempty and the induced candidate-point set Z(s0,t0)\mathcal{Z}(s_0,t_0) may be the whole surface (Remark 4.5 of the source).

References

  1. Sonia Pérez-Díaz, J. Rafael Sendra, Carlos Villarino, 'A First Approach towards Normal Parametrizations of Algebraic Surfaces', International Journal of Algebra and Computation 20(8):977-990, 2010. DOI: 10.1142/S0218196710005972, https://doi.org/10.1142/S0218196710005972
  2. Xiao-Shan Gao, Shang-Ching Chou, 'On the Normal Parameterization of Curves and Surfaces', International Journal of Computational Geometry & Applications 1(2):125-136, 1991. DOI: 10.1142/S0218195991000116, https://doi.org/10.1142/S0218195991000116
  3. J. Rafael Sendra, 'Normal Parametrizations of Algebraic Plane Curves', Journal of Symbolic Computation 33(6):863-885, 2002. DOI: 10.1006/jsco.2002.0538, https://doi.org/10.1006/jsco.2002.0538
  4. J. Rafael Sendra, David Sevilla, Carlos Villarino, 'Some Results on the Surjectivity of Surface Parametrizations', in Computer Algebra and Polynomials, Lecture Notes in Computer Science 8942, pp. 192-203, Springer, 2015. DOI: 10.1007/978-3-319-15081-9_11, https://doi.org/10.1007/978-3-319-15081-9_11
  5. Jorge Caravantes, J. Rafael Sendra, David Sevilla, Carlos Villarino, 'On the existence of birational surjective parametrizations of affine surfaces', Journal of Algebra 501:206-214, 2018. DOI: 10.1016/j.jalgebra.2017.12.028, https://doi.org/10.1016/j.jalgebra.2017.12.028 (arXiv:1706.00492)
  6. Edoardo Ballico, Claudio Fontanari, 'On the existence of finite surjective parametrizations of affine surfaces', Journal of Algebra 601:101-104, 2022. DOI: 10.1016/j.jalgebra.2022.02.021, https://doi.org/10.1016/j.jalgebra.2022.02.021 (arXiv:2107.11794)
  7. J. Rafael Sendra, David Sevilla, Carlos Villarino, 'Covering of surfaces parametrized without projective base points', in Proceedings of ISSAC 2014, pp. 375-380, ACM, 2014. DOI: 10.1145/2608628.2608635, https://doi.org/10.1145/2608628.2608635
  8. Jorge Caravantes, J. Rafael Sendra, David Sevilla, Carlos Villarino, 'Covering Rational Surfaces with Rational Parametrization Images', Mathematics 9(4):338, 2021. DOI: 10.3390/math9040338, https://doi.org/10.3390/math9040338 (arXiv:2101.07011)
  9. Jorge Caravantes, J. Rafael Sendra, David Sevilla, Carlos Villarino, 'A note about rational surfaces as unions of affine planes', Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 117, article 69, 2023. DOI: 10.1007/s13398-023-01401-1, https://doi.org/10.1007/s13398-023-01401-1
  10. Jorge Caravantes, Sonia Pérez-Díaz, J. Rafael Sendra, 'Birational reparameterizations of surfaces', Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 118, article 109, 2024. DOI: 10.1007/s13398-024-01609-9, https://doi.org/10.1007/s13398-024-01609-9 (preprint arXiv:2211.07450, 2022, https://arxiv.org/abs/2211.07450)
  11. Jorge Caravantes, J. Rafael Sendra, David Sevilla, Carlos Villarino, 'Sufficient conditions for the surjectivity of radical curve parametrizations', Journal of Algebra 640:129-146, 2024. DOI: 10.1016/j.jalgebra.2023.11.004, https://doi.org/10.1016/j.jalgebra.2023.11.004
  12. Sebastian Falkensteiner, J. Rafael Sendra, 'Rationality and parametrizations of algebraic curves under specializations', Journal of Algebra 659:698-744, 2024. DOI: 10.1016/j.jalgebra.2024.07.009, https://doi.org/10.1016/j.jalgebra.2024.07.009