9 papers
On the inhomogeneous Vinogradov system
Julia Brandes, Kevin Hughes
We show that the system of equations \begin{align*} \sum_{i=1}^s (x_i^j-y_i^j) = a_j \qquad (1 \le j \le k) \end{align*} has appreciably fewer solutions in the subcritical range $s…
Two-dimensional Weyl sums failing square-root cancellation along lines
Julia Brandes, Igor E. Shparlinski
We show that a certain two-dimensional family of Weyl sums of length takes values as large as on almost all linear slices of the unit torus, contradicting a wi…
The density of rational lines on hypersurfaces: A bihomogeneous perspective
Julia Brandes
Let be a non-singular homogeneous polynomial of degree in variables. We give an asymptotic formula of the pairs of integer points with $|\mathb…
The Hasse principle for diagonal forms restricted to lower-degree hypersurfaces
Julia Brandes, Scott T. Parsell
We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form of degree and one general form of degree , where is smaller than . B…
On generating functions in additive number theory, II: Lower-order terms and applications to PDEs
Julia Brandes, Scott T. Parsell, Konstantinos Poulias +2
We obtain asymptotics for sums of the form involving lower order main terms. As an application, we show that for almost all one…
Optimal mean value estimates beyond Vinogradov's mean value theorem
Julia Brandes, Trevor D. Wooley
We establish improved mean value estimates associated with the number of integer solutions of certain systems of diagonal equations, in some instances attaining the sharpest conjec…