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Computing the dimension of a real algebraic set
arXiv - CS - Symbolic Computation Pub Date : 2021-05-21 , DOI: arxiv-2105.10255
Piere Lairez, Mohab Safey El Din

Let $V$ be the set of real common solutions to $F = (f_1, \ldots, f_s)$ in $\mathbb{R}[x_1, \ldots, x_n]$ and $D$ be the maximum total degree of the $f_i$'s. % The We design an algorithm which on input $F$ computes the dimension of $V$. Letting $L$ be the evaluation complexity of $F$ and $s=1$, it runs using $O^\sim \big (L D^{n(d+3)+1}\big )$ arithmetic operations in $\mathbb{Q}$ and at most $D^{n(d+1)}$ isolations of real roots of polynomials of degree at most $D^n$. Our algorithm depends on the \emph{real} geometry of $V$; its practical behavior is more governed by the number of topology changes in the fibers of some well-chosen maps. Hence, the above worst-case bounds are rarely reached in practice, the factor $D^{nd}$ being in general much lower on practical examples. We report on an implementation showing its ability to solve problems which were out of reach of the state-of-the-art implementations.

中文翻译:

计算实数代数集的维数

假设$ V $是$ \ mathbb {R} [x_1,\ ldots,x_n] $和$ D $中$ F =(f_1,\ ldots,f_s)$的实际公共解集。 $ f_i $的。%我们设计了一种算法,该算法在输入$ F $时计算$ V $的维数。假设$ L $为$ F $的评估复杂度,而$ s = 1 $,则它在$中使用$ O ^ \ sim \ big(LD ^ {n(d + 3)+1} \ big)$算术运算运行\ mathbb {Q} $和最多$ D ^ {n(d + 1)} $个最多$ D ^ n $的多项式的实根的隔离。我们的算法取决于$ V $的\ emph {real}几何;它的实际行为更多地取决于某些精心选择的图的光纤中拓扑变化的数量。因此,在实践中很少达到上述最坏情况的界限,在实际示例中,系数$ D ^ {nd} $通常要低得多。
更新日期:2021-05-24
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