The notion of correct test sequence was introduced in [27]. It has been widely used to design probabilistic algorithms for Polynomial Identity Testing (PIT). In this manuscript we study foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent in the mathematical literature: As hitting sets in PIT, in Function Identity Testing, as norming sets ([1]) or in Reproducing Kernel Hilbert Spaces context. We generalize the main statement of [27] proving that short correct test sequences for lists of polynomials are densely distributed in any constructible set of accurate co–dimension and degree. In order to prove this result we introduce and develop the theory of degree of constructible sets, generalizing [25] and proving two Bezout's Inequalities for two different notions of degree. We exhibit the strength and limitations of correct test sequences with a randomized efficient algorithm for the “Suite Sécante” Problem. Our algorithm generalizes [27], admitting a bigger class of sampling sets, also proving that PIT is in ${\mathbf{RP}}_K$. We reformulate, prove and generalize two results of the Polynomial Method: Dvir's exponential lower bounds for Kakeya sets and Alon's Combinatorial Nullstellensatz.

在[27]中引入了正确测试序列的概念。它已被广泛用于设计多项式身份测试(PIT) 的概率算法。在这份手稿中，我们研究了这个概念的基础和概括。我们证明了正确的测试序列在数学文献中几乎无处不在：作为PIT 中的命中集，在函数身份测试中，作为规范集([1]) 或在复制内核希尔伯特空间上下文中。我们概括了 [27] 的主要陈述，证明多项式列表的短正确测试序列密集分布在任何可构造的准确协维和度集合中。为了证明这个结果，我们引入并发展了可构造集的度理论，概括[25]并证明了两个不同度概念的两个 Bezout 不等式。我们使用针对“Suite Sécante”问题的随机有效算法展示了正确测试序列的优势和局限性。我们的算法对 [27] 进行了概括，承认了更大类别的采样集，也证明了 PIT 在${\mathbf{RP}}_{钾}$. 我们重新表述、证明和概括多项式方法的两个结果：Kakeya 集的 Dvir 指数下界和Alon 的组合 Nullstellensatz。