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Computing real radicals and S-radicals of polynomial systems
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 f=(f1,,fs) be a sequence of polynomials in Q[X1,,Xn] of maximal degree D and VCn be the algebraic set defined by f and r be its dimension. The real radical fre associated to f is the largest ideal which defines the real trace of V. When V is smooth, we show that fre, has a finite set of generators with degrees bounded by degV. Moreover, we present a probabilistic algorithm of complexity (snDn)O(1) to compute the minimal primes of fre. When V is not smooth, we give a probabilistic algorithm of complexity sO(1)(nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set VRn.

Let (g1,,gp) in Q[X1,,Xn] and S be the basic closed semi-algebraic set defined by g10,,gp0. The S-radical of f, which is denoted by fS, is the ideal associated to the Zariski closure of VS. We give a probabilistic algorithm to compute rational parametrizations of all irreducible components of that Zariski closure, hence encoding fS. Assuming now that D is the maximum of the degrees of the fi's and the gi's, this algorithm runs in time 2p(s+p)O(1)(nD)O(rn2r). 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

F=F1个Fs 是以下多项式的序列 [X1个Xñ]最大度DVCñ是由f定义的代数集,r是维数。真正的激进分子F[RËf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F[RË,具有度为的有限生成器集 V。此外,我们提出了一种概率概率算法sñdñØ1个 计算的最小素数 F[RË。当V不平滑时,我们给出一个概率的复杂度算法sØ1个ñdØñ[R2[R 计算实代数集中所有不可约成分的有理参数化 V[Rñ

G1个Gp[X1个Xñ]S是由定义的基本封闭半代数集G1个0Gp0。该Ş -基团的F,由表示 F小号是与Zariski封闭相关的理想选择 V小号。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F小号。假设D是该度数的最大值F一世G一世的,此算法及时运行 2ps+pØ1个ñdØ[Rñ2[R。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。

更新日期:2019-11-04
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