On integer solutions of Parsell–Vinogradov systems

Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems

“…For moment manifolds, the transversality condition was reduced to a conjecture in linear algebra in [BDGuo, Conjecture 3.1] (in the case D = S k corresponding to Parsell-Vinogradov systems, a similar reduction can be made for arbitrary down-sets D, see Definition 4.4). For Parsell-Vinogradov systems this conjecture was verified in [GZh19] using an extension of the Schwartz-Zippel lemma. Our first contribution is the verification of the conjecture for the wider class of sets D in Theorem 1.1, see Section 4.…”

“…In the remaining part of Section 4 we estimate the rank on the right-hand side of (4.3) from below for general d by refining the arguments in [GZh19]. 4.2.…”

“…. = k d = k is the Parsell-Vinogradov case treated in [GZh19]. In this case we have K j,k := K(D(k 1 = k, .…”

“…Non-transverse contributions come from neighborhoods of low degree varieties in [0, 1] d and are handled using the inductive hypothesis with a lower d. In the case of the paraboloid in [BD15], these low degree subvarieties were hyperplanes. Higher degree varieties first appeared in [BD16b]; our treatment mostly follows [GZh19].…”

“…A similar idea (in the case d = 1) independently appeared in a blog post by Terence Tao, from which we adopted the definition (3.22). Putting all estimates in the induction on scales procedure on an equal footing allows us to replace a host of ad hoc calculations of [GZh19] by Theorem 3.3 that describes the right Perron-Frobenius eigenvector of the matrix that contains all essential information about inequalities used. Theorem 3.3 holds for arbitrary down-sets D (see Definition 4.4), thus reducing possible generalizations of Theorem 1.1 to the verification of the transversality condition.…”