Abstract Grobner basis methods are used to solve systems of polynomial equations over finite fields, but their complexity is poorly understood. In this work an upper bound on the time complexity of constructing a Grobner basis according to a total degree monomial ordering and finding a solution of a system is proved. A key parameter in this estimate is the degree of regularity of the leading forms of the polynomials. Therefore, we provide an upper bound to the degree of regularity for a sufficiently overdetermined system of forms of the same degree over any finite field. The bound holds for almost all polynomial system and depends only on the number of variables, the number of polynomials, and the degree. Our results imply that almost all sufficiently overdetermined systems of polynomial equations of the same degree are solvable in polynomial time.

中文翻译：

---

有限域上Macaulay矩阵的概率分析和构造Gröbner基的复杂性

摘要 Grobner 基方法用于求解有限域上的多项式方程组，但对其复杂性知之甚少。在这项工作中，证明了根据总度单项式排序构造 Grobner 基并找到系统解的时间复杂度的上限。此估计中的一个关键参数是多项式的主要形式的规律性程度。因此，我们为在任何有限域上具有相同度数的形式的充分超定系统提供了规则度的上限。该界限几乎适用于所有多项式系统，并且仅取决于变量的数量、多项式的数量和次数。我们的结果意味着几乎所有相同阶多项式方程的充分超定系统都可以在多项式时间内求解。