Let $\mathbf{f} = \left(f_1, \dots, f_p\right) $ be a polynomial tuple in $\mathbb{Q}[z_1, \dots, z_n]$ and let $d = \max_{1 \leq i \leq p} \deg f_i$. We consider the problem of computing the set of asymptotic critical values of the polynomial mapping, with the assumption that this mapping is dominant, $\mathbf{f}: z \in \mathbb{K}^n \to (f\_1(z), \dots, f\_p(z)) \in \mathbb{K}^p$ where $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$. This is the set of values $c$ in the target space of $\mathbf{f}$ such that there exists a sequence of points $(\mathbf{x}_i)_{i\in \mathbb{N}}$ for which $\mathbf{f}(\mathbf{x}_i)$ tends to $c$ and $\|\mathbf{x}_i\| \kappa {\rm d} \mathbf{f}(\mathbf{x}_i))$ tends to $0$ when $i$ tends to infinity where ${\rm d} \mathbf{f}$ is the differential of $\mathbf{f}$ and $\kappa$ is a function measuring the distance of a linear operator to the set of singular linear operators from $\mathbb{K}^n$ to $\mathbb{K}^p$. Computing the union of the classical and asymptotic critical values allows one to put into practice generalisations of Ehresmann's fibration theorem. This leads to natural and efficient applications in polynomial optimisation and computational real algebraic geometry. Going back to previous works by Kurdyka, Orro and Simon, we design new algorithms to compute asymptotic critical values. Through randomisation, we introduce new geometric characterisations of asymptotic critical values. This allows us to dramatically reduce the complexity of computing such values to a cost that is essentially $O(d^{2n(p+1)})$ arithmetic operations in $\mathbb{Q}$. We also obtain tighter degree bounds on a hypersurface containing the asymptotic critical values, showing that the degree is at most $p^{n-p+1}(d-1)^{n-p}(d+1)^{p}$. Next, we show how to apply these algorithms to unconstrained polynomial optimisation problems and the problem of computing sample points per connected component of a semi-algebraic set defined by a single inequality/inequation. We report on the practical capabilities of our implementation of this algorithm. It shows how the practical efficiency surpasses the current state-of-the-art algorithms for computing asymptotic critical values by tackling examples that were previously out of reach.

中文翻译：

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多项式图的渐近临界值的计算及应用

令$ \ mathbf {f} = \ left（f_1，\ dots，f_p \ right）$是$ \ mathbb {Q} [z_1，\ dots，z_n] $中的多项式元组，令$ d = \ max_ {1 \ leq i \ leq p} \ deg f_i $。我们考虑计算多项式映射的渐近临界值集合的问题，并假设该映射是主要映射，$ \ mathbf {f}：z \ in \ mathbb {K} ^ n \ to（f \ _1（ z），\ dots，f \ _p（z））\ in \ mathbb {K} ^ p $中，其中$ \ mathbb {K} $是$ \ mathbb {R} $或$ \ mathbb {C} $。这是目标空间$ \ mathbf {f} $中的值$ c $的集合，因此在\ mathbb {N}} $中存在一系列点$（\ mathbf {x} _i）_ {i \其中$ \ mathbf {f}（\ mathbf {x} _i）$倾向于$ c $和$ \ | \ mathbf {x} _i \ | \ kappa {\ rm d} \ mathbf {f}（\ mathbf {x} _i））$在$ i $趋于无穷大的情况下倾向于$ 0 $，其中$ {\ rm d} \ mathbf {f} $是$ \ mathbf {f} $和$ \ kappa $是一个函数，用于测量从$ \ mathbb {K} ^ n $到$ \ mathbb {K} ^ p $的线性算子到一组奇异线性算子的距离。计算经典临界值和渐近临界值的并集可以使人们将Ehresmann纤维定理的一般化实践付诸实践。这导致在多项式优化和计算实数代数几何中自然而有效的应用。回到Kurdyka，Orro和Simon的先前作品，我们设计了新的算法来计算渐近临界值。通过随机化，我们引入渐近临界值的新几何特征。这使我们能够极大地降低计算此类值的复杂度，其成本基本上是$ \ mathbb {Q} $中的$ O（d ^ {2n（p + 1）}）$算术运算。我们还在包含渐近临界值的超曲面上获得了更严格的度界，表明该度最大为$ p ^ {n-p + 1}（d-1）^ {np}（d + 1）^ {p} $。接下来，我们展示如何将这些算法应用于无约束的多项式优化问题，以及如何计算由单个不等式/不等式定义的半代数集的每个连接部分的采样点问题。我们报告了该算法实现的实际功能。