A promenade through correct test sequences I: Degree of constructible sets, Bézout's Inequality and density

Abstract

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 RPK. We reformulate, prove and generalize two results of the Polynomial Method: Dvir's exponential lower bounds for Kakeya sets and Alon's Combinatorial Nullstellensatz.