Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Linear algebra algorithms for divisors on an algebraic curve

Author: Kamal Khuri-Makdisi
Journal: Math. Comp. 73 (2004), 333-357
MSC (2000): Primary 11Y16, 14Q05, 14H40, 11G20
Published electronically: July 7, 2003
MathSciNet review: 2034126
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess’ 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess’ algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and $C_{ab}$ curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of $O(g^2)$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11Y16, 14Q05, 14H40, 11G20

Retrieve articles in all journals with MSC (2000): 11Y16, 14Q05, 14H40, 11G20

Additional Information

Kamal Khuri-Makdisi
Affiliation: Mathematics Department and Center for Advanced Mathematical Sciences, American University of Beirut, Bliss Street, Beirut, Lebanon
MR Author ID: 610136

Received by editor(s): November 7, 2001
Received by editor(s) in revised form: March 29, 2002
Published electronically: July 7, 2003
Article copyright: © Copyright 2003 American Mathematical Society