A modular method to compute the rational univariate representation of zero-dimensional ideals
Given an ideal \(I \subseteq K[x _1,\ldots,x _n]\), it is a difficult task to determine the set of zeros of \(I\), even if \(I\) is known to be zero-dimensional. An important tool for this is the use of Gröbner bases, which by the so-called shape lemma often have a form suitable for finding zeros. The authors propose a new method to find a rational univariate representation (RUR) for \(I\), as defined by \textit{F. Rouillier} [Appl. Algebra Eng. Commun. Comput. 9, No.~5, 433-461 (1999; Zbl 0932.12008)]. This is another ideal basis which has the form \[ \{f(u),g _1(u) x _1 - h _1(u),\ldots,g _n(u) x _n - h _n(u)\} \] with \(u \in K[x _1,\ldots,x _n]\) a ``separating element'' and \(f,g _i,h _i\) univariate polynomials. The advantage over a Gröbner basis is that an RUR often has smaller coefficients. The new method applies to the case where the ground field \(K\) is \(\mathbb Q\) and uses reduction modulo a prime and Hensel lifting.
scientific article; zbMATH DE number 1369547