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Register a new userUnirational Differential Curves and Differential Rational Parametrizations
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In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore, for linear differential curves, we give an algorithm to determine whether an implicitly given linear differential curve is unirational and, in the affirmative case, to compute a proper differential rational parametrization for the differential curve.
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In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are $G$-equivalent. We prove that for any $G$-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group and some quantities depending on a choice of classifying invariants. For the full projective group, as well as for its affine, special affine and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve. We show that this generic degree is the sharp upper bound.
We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle $S^1$ admits a unique smooth rational real quasi-projective model up to $mathbb{S}^1$-equivariant birational diffeomorphism.
We show that the results we had obtained on diagonals of nine and ten parameters families of rational functions using creative telescoping, yielding modular forms expressed as pullbacked $ _2F_1$ hypergeometric functions, can be obtained, much more efficiently, calculating the $ j$-invariant of an elliptic curve canonically associated with the denominator of the rational functions. In the case where creative telescoping yields pullbacked $ _2F_1$ hypergeometric functions, we generalize this result to other families of rational functions in three, and even more than three, variables. We also generalise this result to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, foliation in elliptic curves. We also extend these results to rational functions in three variables when the denominator is associated with a {em genus-two curve such that its Jacobian is a split Jacobian} corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked $ _2F_1$ hypergeometric solutions, the denominator corresponding to an algebraic variety being not simply foliated in elliptic curves, but having a selected elliptic curve in the variety explaining the pullbacked $ _2F_1$ hypergeometric solution.
In a joint work with N. Mok in 1997, we proved that for an irreducible representation $G subset {bf GL}(V),$ if a holomorphic $G$-structure exists on a uniruled projective manifold, then the Lie algebra of $G$ has nonzero prolongation. We tried to generalize this to an arbitrary connected algebraic subgroup $G subset {bf GL}(V)$ and a complex manifold containing an immersed rational curve, but the proposed proof had a flaw.
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.
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