In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout’s theorem, IEEE Trans. Inform. Theory 46(2) (2000) 635–641, On (or in) Dick Blahut’s ‘footprint’, Codes, Curves Signals (1998) 3–9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.

中文翻译：

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在有限网格上具有多个根的多元多项式

在本文中，我们考虑了有限网格上的多元多项式的根。当给出关于固定单项式排序的前导单项式的信息时，足迹绑定 [足迹或广义 Bezout 定理，IEEE Trans。通知。理论 46(2) (2000) 635–641，在（或在）Dick Blahut 的“足迹”上，代码,曲线信号(1998) 3-9] 为我们提供了根数的上限，这个界限很明确，因为它总是可以通过平凡多项式获得，该多项式是常数乘以由变量组成的项的适当组合的乘积减去一个常数。与一种可变情况相比，存在不属于上述形式的多变量多项式获得足迹界限。这甚至包括不可约多项式。本文的目的是确定只有提到的平凡多项式才能达到界限的一大类多项式，这意味着要搜索具有最大根数的其他多项式，必须在该类之外寻找。