论文标题
通过正确的测试序列I：可构造度的程度，Bézout的不平等和密度的长廊
A promenade through Correct Test Sequences I: Degree of constructible sets, Bézout's Inequality and density
论文作者
论文摘要
在Heintz-Schnorr（1982）中，作者介绍了正确的测试序列的概念，从那时起，它被广泛用于设计概率算法以进行多项式相等测试。该手稿的目的是研究该概念的基础和概括。我们表明，正确的测试序列几乎是无处不在的，并且在数学文献中以许多不同的形式出现：作为功能身份测试的身份序列，作为Banach代数领域的规范集或在再现核Hilbert Space的背景下作为样本。作为主要结果，我们概括了Heintz-Schnorr（1982）的主要陈述，证明确实存在了可构造的多项式列表集的简短正确测试序列，并且在任何可构造的准确维度和程度集中都密集分布。用于证明这一结果的主要工具是我们在本手稿中介绍和开发的可构造集理论，概括了Heintz（1983）的结果，并证明了两种不同程度概念的Bezout的不平等现象。我们提出了一个$ {\ bf bpp} _k $算法以展示正确的测试序列的功能，该算法决定只要在一些合理的点上评估输入列表，就决定了多项式列表是否是暂时序列。我们显示了正确的测试序列与demillo-lipton-schwartz-Zippel概率测试之间的差异，并且我们重新制定，证明和概括了多项式方法的两个众所周知的结果：我们证明了DVIR的指数下限是Kakeya的kakeya sets s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s semos semos and allontorsate序列的下层下部下限级别级的下限均来自级的差异。
In Heintz-Schnorr (1982), the authors introduced the notion of correct test sequence and since then it has been widely used to design probabilistic algorithms for Polynomial Equality Test. The aim of this manuscript is to study the foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent and appear in many different forms in the mathematical literature: As identity sequences for Function Identity Test, as norming sets in the field of Banach algebras or as samples in the context of Reproducing Kernel Hilbert Spaces. As main outcome, we generalize the main statement of Heintz-Schnorr (1982) proving that short correct test sequences for constructible sets of lists of polynomials do exist and are densely distributed in any constructible set of accurate dimension and degree. The main tool used to prove this result is the theory of degree of constructible sets, which we introduce and develop in this manuscript, generalizing the results of Heintz (1983) and proving two Bezout's Inequalities for two different notions of degree. We present a ${\bf BPP}_K$ algorithm to exhibit the power of correct test sequences, this algorithm decides whether a list of polynomials is a secant sequence by just evaluating the input list at some well-suited points. We show the differences between correct test sequences and Demillo-Lipton-Schwartz-Zippel probabilistic tests and we reformulate, prove and generalize two well-known results of the Polynomial Method: We prove Dvir's exponential lower bounds for Kakeya sets from lower bounds for the length of correct test sequences and generalize Alon's Combinatorial Nullstellensatz.
