This short paper has two goals. First, explaining a simple procedure (which is essentially folklore) that, sometimes, makes it possible to obtain a formula for the number of solutions to a system of multivariate polynomial inequalities over a finite field. Second, applying that procedure to prove a formula for the number of contiguous superregular $4 \times 4$ matrices over a finite field. The formula was previously conjectured by Appuswamy, Bazzani, Connelly, Ekaireb, Congero, and Zeger [Probability of super-regular matrices and MDS codes over finite fields, arXiv:2603.20983]. In addition, the same procedure is used to provide formulas for the number of contiguous superregular $3 \times 4$, $3 \times 5$, and $3 \times 6$ matrices over a finite field.

翻译：这篇短文有两个目标。首先，解释一个简单的过程（本质上是民间知识），该过程有时能够得出有限域上多元多项式不等式系统解数量的公式。其次，应用该过程证明有限域上连续超正则 $4 \times 4$ 矩阵数量的公式。该公式此前由 Appuswamy, Bazzani, Connelly, Ekaireb, Congero 和 Zeger 提出猜想 [Super-regular matrices 和 MDS 码在有限域上的概率, arXiv:2603.20983]。此外，相同过程还被用于提供有限域上连续超正则 $3 \times 4$、$3 \times 5$ 和 $3 \times 6$ 矩阵数量的公式。