Computation of unirational fields

Computation of unirational fields (extended abstract)

In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner bases theory. Our algorithm also requires computing computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated K-algebras.

Unirational fields of transcendence degree one and functional decomposition

In this paper we present an algorithm to compute all unirational fields of transcendence degree one containing a given finite set of multivariate rational functions. In particular, we provide an algorithm to decompose a multivariate rational function f of the form f=g(h), where g is a univariate rational function and h a multivariate one.

有限域上参数理想的根计算

On the Radical Computation of Parametric Ideals over Finite Fields

该研究将Matsumoto的多项式理想根计算算法推广到参数情形，利用有限域上的综合格罗比纳系统，提出有限域上参数理想的根计算算法。

Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields.

On the complexity of class group computations for large degree number fields

In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity analysis and identify the optimal parameters to reduce the runtime. We make use of the classes $\mathcal D$ defined in [GJ16] for classifying the fields according to the size of the extension degree and prove that they enable to describe all the number fields.

Improvements in the computation of ideal class groups of imaginary quadratic number fields

We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear algebra phase. We achieve a significant speed-up and are able to compute ideal class groups with discriminants of 110 decimal digits in less than a week.