We present a probabilistic Las Vegas algorithm for solving sufficiently generic square polynomial systems over finite fields. We achieve a nearly quadratic running time in the number of solutions, for densely represented input polynomials. We also prove a nearly linear bit complexity bound for polynomial systems with rational coefficients. Our results are obtained using the combination of the Kronecker solver and a new improved algorithm for fast multivariate modular composition.

中文翻译：

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多项式系统求解的复杂度指数

我们提出一种概率拉斯维加斯算法，用于求解有限域上的足够通用的平方多项式系统。对于密集表示的输入多项式，我们在解决方案数量上实现了接近二次的运行时间。我们还证明了具有有理系数的多项式系统的近似线性位复杂度界限。我们的结果是通过结合Kronecker求解器和新的改进算法进行快速多元模块化组合而获得的。