稀疏多项式的整除与整除性检验
On exact division and divisibility testing for sparse polynomials
摘要：
已知没有多项式时间算法来测试稀疏多项式G是否整除另一稀疏多项式$F$。虽然计算商Q=F <$G可以在多项式时间内相对于F，G和Q的稀疏性完成，但这还不足以得到一般的多项式时间整除性测试。实际上，商Q的稀疏性可以指数地大于F和G的稀疏性。在商的稀疏性#Q是多项式的有利情况下，计算Q的最佳已知算法在复杂度中具有非线性因子#G#Q，这不是最佳的。在这项工作中，我们对这个问题的两个方面感兴趣。首先，我们提出了一个新的随机算法，计算商的两个稀疏多项式的除法是准确的。它的复杂性在F、G和Q的稀疏性下是拟线性的。我们的方法依赖于稀疏插值，它适用于任何有限域或整数环。然后，作为更快的整除性测试的一步，我们提供了一个新的多项式时间算法时，除数有一个特定的形状。更准确地说，我们减少了问题，找到一个多项式S，使QS是稀疏的和测试整除S可以在多项式时间。我们发现了除数G中的一些结构模式，我们可以有效地计算这样的多项式~S。
英文摘要：
No polynomial-time algorithm is known to test whether a sparse polynomial G divides another sparse polynomial $F$. While computing the quotient Q=F quo G can be done in polynomial time with respect to the sparsities of F, G and Q, this 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 field or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial-time algorithm when the divisor has a specific shape. 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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