4,942 research outputs found
Rational solutions of pairs of diagonal equations, one cubic and one quadratic
We obtain an essentially optimal estimate for the moment of order 32/3 of the
exponential sum having argument . Subject to modest local
solubility hypotheses, we thereby establish that pairs of diagonal Diophantine
equations, one cubic and one quadratic, possess non-trivial integral solutions
whenever the number of variables exceeds 10
Mean value estimates for odd cubic Weyl sums
We establish an essentially optimal estimate for the ninth moment of the
exponential sum having argument . The first substantial
advance in this topic for over 60 years, this leads to improvements in
Heath-Brown's variant of Weyl's inequality, and other applications of
Diophantine type
On Waring's problem: some consequences of Golubeva's method
We investigate sums of mixed powers involving two squares, two cubes, and
various higher powers, concentrating on situations inaccessible to the
Hardy-Littlewood method
Solvable points on smooth projective varieties
We establish that smooth, geometrically integral projective varieties of
small degree are not pointless in suitable solvable extensions of their field
of definition, provided that this field is algebraic over .Comment: 11 page
Multigrade efficient congruencing and Vinogradov's mean value theorem
We develop a multigrade enhancement of the efficient congruencing method to
estimate Vinogradov's integral of degree for moments of order , thereby
obtaining near-optimal estimates for . There are
numerous applications. In particular, when is large, the anticipated
asymptotic formula in Waring's problem is established for sums of th
powers of natural numbers whenever . The asymptotic formula is also
established for sums of fifth powers.Comment: 48pp; modest revisions in light of referee comment
Translation invariance, exponential sums, and Waring's problem
We describe mean value estimates for exponential sums of degree exceeding 2
that approach those conjectured to be best possible. The vehicle for this
recent progress is the efficient congruencing method, which iteratively
exploits the translation invariance of associated systems of Diophantine
equations to derive powerful congruence constraints on the underlying
variables. There are applications to Weyl sums, the distribution of polynomials
modulo 1, and other Diophantine problems such as Waring's problem.Comment: Submitted to Proceedings of the ICM 201
Approximating the main conjecture in Vinogradov's mean value theorem
We apply multigrade efficient congruencing to estimate Vinogradov's integral
of degree for moments of order , establishing strongly diagonal
behaviour for . In particular,
as , we confirm the main conjecture in Vinogradov's mean
value theorem for 100% of the critical interval .Comment: arXiv admin note: text overlap with arXiv:1310.844
Discrete Fourier restriction via efficient congruencing: basic principles
We show that whenever , then for any complex sequence , one has Bounds for
the constant in the associated periodic Strichartz inequality from to
of the conjectured order of magnitude follow, and likewise for the
constant in the discrete Fourier restriction problem from to ,
where . These bounds are obtained by generalising the efficient
congruencing method from Vinogradov's mean value theorem to the present
setting, introducing tools of wider application into the subject.Comment: 37 page
On Waring's problem for intermediate powers
Let denote the least number such that every sufficiently large
natural number is the sum of at most positive integral th powers. We
show that , , , , , , , , , .Comment: 7 page
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