Quaternion rational surfaces.

*(English)*Zbl 1445.14082Let \(f(s,u)\) and \(g(t,v)\) two one-to-one real projective parametrizations of rational space curves. The quaternion multiplication \(f(s,u)*g(t,v)=h(s,u;t,v)\) yields an one-to-one parametrization of a real rational surface called quaternion rational surface. The surface is assumed to have no base points. Quaternions have the important property that every unit represents a rotation and the multiplication of two quaternions corresponds to the composition of rotations; this allows changing quaternion rational surfaces (location, orientation and shape) by manipulating the two rational space curves that generate the surface. As the authors say in the abstract, since the structure of the graded minimal free resolution of a rational surface is, in general, unknown, the main goal of the paper is, under certain hypotheses, to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. They give explicit formulas for the maps of these graded minimal free resolutions by using \(\mu\)-bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. Necessary and sufficient conditions for a real rational surface to be a quaternion rational surface are also provided. Finally, the authors find the relationsheep between the ideal generated by the parametrization and the ideal generated by the moving planes; they prove that the ideal generated by the first syzygy module expressed in terms of moving planes is the same as the ideal generated by the parametrization in the affine ring.

Reviewer: Carlos Hermoso Ortíz (Madrid)

##### MSC:

14Q05 | Computational aspects of algebraic curves |

14Q10 | Computational aspects of algebraic surfaces |

13D02 | Syzygies, resolutions, complexes and commutative rings |

##### References:

[1] | N. K. Bose, Multidimensional systems theory and applications, 2nd ed., Springer, 1995. |

[2] | N. Bourbaki, Commutative algebra, Chapters I-VII, Springer, 1989. · Zbl 0666.13001 |

[3] | J. G. Broida and S. G. Williamson, A comprehensive introduction to linear algebra, Addison-Wesley, 1989. · Zbl 0683.15001 |

[4] | F. Chen and W. Wang, “Revisiting the \(\mu \)-basis of a rational ruled surface”, J. Symbolic Comput. 36:5 (2003), 699-716. · Zbl 1037.14014 |

[5] | F. Chen, D. Cox, and Y. Liu, “The \(\mu \)-basis and implicitization of a rational parametric surface”, J. Symbolic Comput. 39:6 (2005), 689-706. · Zbl 1120.14054 |

[6] | D. A. Cox, “Equations of parametric curves and surfaces via syzygies”, pp. 1-20 in Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000), edited by E. L. Green et al., Contemp. Math. 286, Amer. Math. Soc., 2001. · Zbl 1009.68174 |

[7] | D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, 3rd ed., Springer, 2007. · Zbl 1118.13001 |

[8] | J. Deng, F. Chen, and L. Shen, “Computing \(\mu \)-bases of rational curves and surfaces using polynomial matrix factorization”, pp. 132-139 in ISSAC’05, edited by M. Kauers, ACM, 2005. · Zbl 1360.14141 |

[9] | E. Duarte and H. Schenck, “Tensor product surfaces and linear syzygies”, Proc. Amer. Math. Soc. 144:1 (2016), 65-72. · Zbl 1327.14225 |

[10] | D. Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995. · Zbl 0819.13001 |

[11] | J. W. Hoffman and H. H. Wang, “Castelnuovo-Mumford regularity in biprojective spaces”, Adv. Geom. 4:4 (2004), 513-536. · Zbl 1063.13001 |

[12] | J. E. Mebius, “A matrix-based proof of the quaternion representation theorem for four-dimensional rotations”, preprint, 2005. |

[13] | H. Schenck, A. Seceleanu, and J. Validashti, “Syzygies and singularities of tensor product surfaces of bidegree \((2,1)\)”, Math. Comp. 83:287 (2014), 1337-1372. · Zbl 1286.13013 |

[14] | T. W. Sederberg and F. Chen, “Implicitization Using Moving Curves and Surfaces”, pp. 301-308 in Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, ACM, 1995. |

[15] | K. Shoemake, “Quaternions and \(4{\times} 4\) matrices”, pp. 351-354 in Graphics gems II, edited by J. Arvo, Academic Press, 1991. |

[16] | H. Wang, Equations of parametric surfaces with base points via syzygies, Ph.D. thesis, Louisiana State University, 2003, https://digitalcommons.lsu.edu/gradschool_dissertations/3973. |

[17] | X. Wang and R. Goldman, “Quaternion rational surfaces: rational surfaces generated from the quaternion product of two rational space curves”, Graphical Models 81 (2015), 18-32. |

[18] | S. · Zbl 1132.65011 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.