Let $\mathit{f}=(f_1,\dots ,f_s)$ be a sequence of polynomials in $\mathbb{Q}[X_1,\dots ,X_n]$ of maximal degree D and $V\subset {\mathbb{C}}^n$ be the algebraic set defined by f and r be its dimension. The real radical $\sqrt[re]{〈\mathit{f}〉}$ associated to f is the largest ideal which defines the real trace of V. When V is smooth, we show that $\sqrt[re]{〈\mathit{f}〉}$, has a finite set of generators with degrees bounded by $\mathrm{deg}V$. Moreover, we present a probabilistic algorithm of complexity ${(sn{D}^n)}^{O(1)}$ to compute the minimal primes of $\sqrt[re]{〈\mathit{f}〉}$. When V is not smooth, we give a probabilistic algorithm of complexity $s^{O(1)}{(nD)}^{O(nr{2}^r)}$ to compute rational parametrizations for all irreducible components of the real algebraic set $V\cap {\mathbb{R}}^n$.

Let $(g_1,\dots ,g_p)$ in $\mathbb{Q}[X_1,\dots ,X_n]$ and S be the basic closed semi-algebraic set defined by $g_1\ge 0,\dots ,g_p\ge 0$. The S-radical of $〈\mathit{f}〉$, which is denoted by $\sqrt[S]{〈\mathit{f}〉}$, is the ideal associated to the Zariski closure of $V\cap S$. We give a probabilistic algorithm to compute rational parametrizations of all irreducible components of that Zariski closure, hence encoding $\sqrt[S]{〈\mathit{f}〉}$. Assuming now that D is the maximum of the degrees of the $f_i$'s and the $g_i$'s, this algorithm runs in time $2^p{(s+p)}^{O(1)}{(nD)}^{O(rn{2}^r)}$. Experiments are performed to illustrate and show the efficiency of our approaches on computing real radicals.

让 $\mathit{F}=（F_{\mathrm{1个}}，\dots ，F_s）$ 是以下多项式的序列 $\mathbb{问}[X_{\mathrm{1个}}，\dots ，X_{ñ}]$最大度D和$V\subset {\mathbb{C}}^{ñ}$是由f定义的代数集，r是维数。真正的激进分子$\sqrt[\mathrm{[R}Ë]{〈\mathit{F}〉}$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$\sqrt[\mathrm{[R}Ë]{〈\mathit{F}〉}$，具有度为的有限生成器集 $\mathrm{度}V$。此外，我们提出了一种概率概率算法${（sñd^{ñ}）}^{Ø（\mathrm{1个}）}$ 计算的最小素数 $\sqrt[\mathrm{[R}Ë]{〈\mathit{F}〉}$。当V不平滑时，我们给出一个概率的复杂度算法$s^{Ø（\mathrm{1个}）}{（ñd）}^{Ø（ñ\mathrm{[R}{2}^{\mathrm{[R}}）}$ 计算实代数集中所有不可约成分的有理参数化 $V\cap {\mathbb{[R}}^{ñ}$。

让 $（G_{\mathrm{1个}}，\dots ，G_p）$ 在 $\mathbb{问}[X_{\mathrm{1个}}，\dots ，X_{ñ}]$和S是由定义的基本封闭半代数集$G_{\mathrm{1个}}\ge 0，\dots ，G_p\ge 0$。该Ş -基团的$〈\mathit{F}〉$，由表示 $\sqrt[\mathrm{小号}]{〈\mathit{F}〉}$是与Zariski封闭相关的理想选择 $V\cap \mathrm{小号}$。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$\sqrt[\mathrm{小号}]{〈\mathit{F}〉}$。假设D是该度数的最大值$F_{\mathrm{一世}}$和 $G_{\mathrm{一世}}$的，此算法及时运行 $2^p{（s+p）}^{Ø（\mathrm{1个}）}{（ñd）}^{Ø（\mathrm{[R}ñ2^{\mathrm{[R}}）}$。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。