Journal of Symbolic Computation ( IF 0.6 ) Pub Date : 2019-11-04 , DOI: 10.1016/j.jsc.2019.10.018 Mohab Safey El Din , Zhi-Hong Yang , Lihong Zhi
Let be a sequence of polynomials in of maximal degree D and be the algebraic set defined by f and r be its dimension. The real radical associated to f is the largest ideal which defines the real trace of V. When V is smooth, we show that , has a finite set of generators with degrees bounded by . Moreover, we present a probabilistic algorithm of complexity to compute the minimal primes of . When V is not smooth, we give a probabilistic algorithm of complexity to compute rational parametrizations for all irreducible components of the real algebraic set .
Let in and S be the basic closed semi-algebraic set defined by . The S-radical of , which is denoted by , is the ideal associated to the Zariski closure of . We give a probabilistic algorithm to compute rational parametrizations of all irreducible components of that Zariski closure, hence encoding . Assuming now that D is the maximum of the degrees of the 's and the 's, this algorithm runs in time . Experiments are performed to illustrate and show the efficiency of our approaches on computing real radicals.
中文翻译:
计算多项式系统的实根和S根
让 是以下多项式的序列 最大度D和是由f定义的代数集,r是维数。真正的激进分子与f相关联的是最大的理想值,它定义了V的真实轨迹。当V光滑时,我们证明,具有度为的有限生成器集 。此外,我们提出了一种概率概率算法 计算的最小素数 。当V不平滑时,我们给出一个概率的复杂度算法 计算实代数集中所有不可约成分的有理参数化 。
让 在 和S是由定义的基本封闭半代数集。该Ş -基团的,由表示 是与Zariski封闭相关的理想选择 。我们给出一个概率算法来计算该Zariski闭包的所有不可约成分的有理参数化,从而进行编码。假设D是该度数的最大值和 的,此算法及时运行 。进行实验以说明并显示我们的方法在计算实际部首方面的效率。