Assessing that a sparse polynomial G divides another sparse polynomial F is not yet known to admit a polynomial time algorithm. While computing the quotient Q = F quo G can be done in polynomial time with respect to the sparsities of F, G and Q, it is not yet sufficient to get a polynomial time divisibility test in general. Indeed, the sparsity of the quotient Q can be exponentially larger than the ones of F and G. In the favorable case where the sparsity #Q of the quotient is polynomial, the best known algorithm to compute Q has a non-linear factor #G#Q in the complexity, which is not optimal. In this work, we are interested in the two aspects of this problem. First, we propose a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact. Its complexity is quasi-linear in the sparsities of F, G and Q. Our approach relies on sparse interpolation and it works over any finite fields or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial time algorithm when the divisor has some specific shapes. More precisely, we reduce the problem to finding a polynomial S such that QS is sparse and testing divisibility by S can be done in polynomial time. We identify some structure patterns in the divisor G for which we can efficiently compute such a polynomial S.

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关于稀疏多项式的精确除法和除数检验

尚不知道使用一个稀疏多项式G除以另一个稀疏多项式F即可接纳多项式时间算法。相对于F，G和Q的稀疏性，可以在多项式时间内计算商Q = F quo G，但一般而言，尚不足以得到多项式时间可除性检验。确实，商Q的稀疏度可以比F和G的指数级大。在商稀疏#Q是多项式的有利情况下，最著名的计算Q的算法具有非线性因子#G #Q的复杂度不是最佳的。在这项工作中，我们对这个问题的两个方面都很感兴趣。首先，我们提出一种新的随机算法，当除法精确时，该算法可计算两个稀疏多项式的商。在F，G和Q的稀疏度中，其复杂度是准线性的。我们的方法依赖于稀疏插值，并且可以在任何有限域或整数环上工作。然后，作为迈向更快的除数测试的一步，当除数具有某些特定形状时，我们提供了一种新的多项式时间算法。更准确地说，我们将问题简化为找到多项式S，使得QS稀疏，并且可以在多项式时间内完成由S进行除数的测试。我们在除数G中确定一些结构模式，可以有效地计算这样的多项式S。我们将问题简化为找到多项式S，使得QS稀疏，并且可以在多项式时间内完成用S进行除数的测试。我们在除数G中确定一些结构模式，可以有效地计算这样的多项式S。我们将问题简化为找到多项式S，使得QS稀疏，并且可以在多项式时间内完成用S进行除数的测试。我们在除数G中确定一些结构模式，可以有效地计算这样的多项式S。