In this paper, we propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP). Both of our algorithms work over any finite fields $F_q$ with large characteristic. The first one is a Monte Carlo randomized algorithm. Its arithmetic complexity is linear in the number $T$ of non-zero terms of $f$, in the number $n$ of variables. If $q$ is $O((nTD)^{(1)})$, where $D$ is the partial degree bound, then our algorithm has better complexity than other existing algorithms. The second one is a deterministic algorithm. It has better complexity than existing deterministic algorithms over a field with large characteristic. Its arithmetic complexity is quadratic in $n,T,\log D$, i.e., quadratic in the size of the sparse representation. And we also show that the complexity of our deterministic algorithm is the same as the one of deterministic zero-testing of Bl\"{a}ser et al. for the polynomial given by an SLP over finite field (for large characteristic).

中文翻译：

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基于导数的稀疏多项式插值

在本文中，我们针对由直线规划（SLP）表示的稀疏多元多项式提出了两种新的插值算法。我们的两种算法都适用于具有大特征的任何有限域 $F_q$。第一个是蒙特卡罗随机算法。它的算术复杂度在 $f$ 的非零项的数量 $T$ 中，在变量的数量 $n$ 中是线性的。如果 $q$ 是 $O((nTD)^{(1)})$，其中 $D$ 是部分度界，那么我们的算法比其他现有算法具有更好的复杂度。第二种是确定性算法。在具有较大特征的领域上，它比现有的确定性算法具有更好的复杂性。它的算术复杂度是 $n,T,\log D$ 的二次方，即稀疏表示大小的二次方。