In the area of symbolic-numerical computation within computer algebra, an interesting question is how “close” a random input is to the “critical” ones. Examples are the singular matrices in linear algebra or the polynomials with multiple roots for Newton's root-finding method. Bounds, sometimes very precise, are known for the volumes over $\mathbb{R}$ or $\mathbb{C}$ of such neighborhoods of the varieties of “critical” inputs; see the references below.

This paper deals with the discrete version of this question: over a finite field, how many points lie in a certain type of neighborhood around a given variety? A trivial upper bound on this number is given by the product (size of the variety) ⋅ (size of a neighborhood of a point). It turns out that this bound is usually asymptotically tight, in particular for the singular matrices, polynomials with multiple roots, and pairs of non-coprime polynomials.

The interesting question then is: for which varieties is this bound not asymptotically tight? We show that these are precisely those that admit a shift, that is, where one absolutely irreducible component of maximal dimension is a shift (translation by a fixed nonzero point) of another such component. Furthermore, the shift-invariant absolutely irreducible varieties are characterized as being cylinders over some base variety.

Computationally, determining whether a given variety is shift-invariant turns out to be intractable, namely NP-hard even in simple cases.

在计算机代数的符号数值计算领域，一个有趣的问题是随机输入与“关键”输入的“接近”程度。示例是线性代数中的奇异矩阵或用于牛顿求根方法的具有多个根的多项式。界限，有时非常精确，以超过$\mathbb{电阻}$ 要么 $\mathbb{C}$各种“关键”投入的社区；请参阅下面的参考资料。

这篇论文讨论了这个问题的离散版本：在有限域上，在给定品种周围的特定类型的邻域中有多少点？这个数字的一​​个简单的上限由乘积（品种的大小）·（一个点的邻域的大小）给出。事实证明，这个界限通常是渐近紧的，特别是对于奇异矩阵、具有多个根的多项式和非互质多项式对。

那么有趣的问题是：对于哪些品种，这个界限不是渐近紧的？我们证明这些正是那些允许偏移的那些，也就是说，最大维数的一个绝对不可约分量是另一个这样的分量的偏移（由一个固定的非零点平移）。此外，移不变绝对不可约变体的特征是在某些基变体上是圆柱体。

在计算上，确定给定的品种是否是平移不变的，结果证明是难以处理的，即 NP-hard 即使在简单的情况下。