In this contribution, we consider a zero-dimensional polynomial system in $n$ variables defined over a field $\mathbb{K}$. In the context of computing a Rational Univariate Representation (RUR) of its solutions, we address the problem of certifying a separating linear form and, once certified, calculating the RUR that comes from it, without any condition on the ideal else than being zero-dimensional. Our key result is that the RUR can be read (closed formula) from lexicographic Groebner bases of bivariate elimination ideals, even in the case where the original ideal that is not in shape position, so that one can use the same core as the well known FGLM method to propose a simple algorithm. Our first experiments, either with a very short code (300 lines) written in Maple or with a Julia code using straightforward implementations performing only classical Gaussian reductions in addition to Groebner bases for the degree reverse lexicographic ordering, show that this new method is already competitive with sophisticated state of the art implementations which do not certify the parameterizations.

翻译：在本文中，我们考虑定义在域$\mathbb{K}$上的$n$元零维多项式系统。在计算其解的有理单变量表示（RUR）的背景下，我们解决了认证分离线性形式的问题，并在认证后计算由此产生的RUR，对理想除了零维性外不附加任何条件。我们的关键结果是：即使原始理想不处于shape位置，RUR也可以从双变量消元理想的字典序Gröbner基中直接读取（闭式公式），从而可以利用与著名的FGLM方法相同的核心来提出一种简单算法。初步实验——无论是用Maple编写的极简代码（约300行），还是用Julia语言实现、仅需在反向字典序Gröbner基基础上执行经典高斯约简的直白代码——均表明：这种新方法已能与不认证参数化的先进成熟实现相媲美。