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An Efficient Method to Compute the Univariate Rational
        Representation for Zero-Dimensional Systems

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Masayuki Noro, Kazuhiro Yokoyama 

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Date:  July 19th (Friday)\\
Time:  10:30-11:00\\

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Abstract
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Computation of Gr\"obner basis with respect to lexicographic order is one of useful methods for
computing zeros of a zero-dimensional ideal numerically, but it is often practically hard because
of swells of intermediate coefficients and results itself. On the other hand, RUR (Rational
Univariate Representation -- formerly known as GSL) represents results as rational expressions
instead of polynomials, and it is empirically known that the coefficients of the results are very
small compared with those of Gr\"obner basis elements. RUR was proposed by M.-E. Alonso et al.,
and methods of RUR computation using symmetric function were presented by F. Rouillier, L.
Gonzalez-Vega and G. Trujillo. Here we propose another method, where main tools are modular
Gr\"obner bases, linear algebra and Hensel lifting. The method is a modification of our modular
change-of-ordering algorithm for Gr\"obner basis computation, and it is easily parallelized. In this
talk, we will present our method, discuss its efficiency and application to prime decomposition of
zero-dimensional radical ideals. 

