Suppose $\mathbb{K}$ is a large enough field and $\mathcal{P} \subset \mathbb{K}^2$ is a fixed, generic set of points which is available for precomputation. We introduce a technique called \emph{reshaping} which allows us to design quasi-linear algorithms for both: computing the evaluations of an input polynomial $f \in \mathbb{K}[x,y]$ at all points of $\mathcal{P}$; and computing an interpolant $f \in \mathbb{K}[x,y]$ which takes prescribed values on $\mathcal{P}$ and satisfies an input $y$-degree bound. Our genericity assumption is explicit and we prove that it holds for most point sets over a large enough field. If $\mathcal{P}$ violates the assumption, our algorithms still work and the performance degrades smoothly according to a distance from being generic. To show that the reshaping technique may have an impact on other related problems, we apply it to modular composition: suppose generic polynomials $M \in \mathbb{K}[x]$ and $A \in \mathbb{K}[x]$ are available for precomputation, then given an input $f \in \mathbb{K}[x,y]$ we show how to compute $f(x, A(x)) \operatorname{rem} M(x)$ in quasi-linear time.

中文翻译：

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具有预计算的通用双变量多点评估、插值和模块化组合

假设 $\mathbb{K}$ 是一个足够大的字段，而 $\mathcal{P} \subset \mathbb{K}^2$ 是一个固定的通用点集，可用于预计算。我们引入了一种称为 \emph{reshaping} 的技术，它允许我们为两者设计拟线性算法：计算输入多项式 $f \in \mathbb{K}[x,y]$ 在 $\ 的所有点的评估数学{P}$; 并计算一个插值 $f \in \mathbb{K}[x,y]$ ，它在 $\mathcal{P}$ 上取指定值并满足输入 $y$-degree 界限。我们的通用性假设是明确的，我们证明它适用于足够大的领域上的大多数点集。如果 $\mathcal{P}$ 违反假设，我们的算法仍然有效，并且性能会根据与泛型的距离平滑下降。为了表明重塑技术可能对其他相关问题产生影响，