18 citations · 42 across the 19 of their papers we have counts for
24 papers
Structure and paucity in affine diagonal systems, I
Julia Brandes, Trevor D. Wooley
Let and . We show that whenever is large and the system \[ x_1^j+x_2^j-y_1^j-y_2^j=h_j\quad (j=1,2,3) \] has more than …
Rational lines on cubic hypersurfaces II
Julia Brandes, Rainer Dietmann, David B. Leep
We show that any rational cubic hypersurface of dimension at least 33 defined over a number field vanishes on a -rational projective line, reducing the previous lower bound…
A minimalist version of the circle method and Diophantine problems over thin sets
Kirsti D. Biggs, Julia Brandes
This paper studies the minimal conditions under which we can establish asymptotic formulae for Waring's problem and other additive problems that may be tackled by the circle method…
Local mean value estimates for Weyl sums
Julia Brandes, Changhao Chen, Igor E. Shparlinski
We obtain new estimates - both upper and lower bounds - on the mean values of the Weyl sums over a small box inside of the unit torus. In particular, we refine recent conjectures o…
Campana points on diagonal hypersurfaces
Francesca Balestrieri, Julia Brandes, Miriam Kaesberg +3
We construct an integral model for counting Campana points of bounded height on diagonal hypersurfaces of degree greater than one, and give an asymptotic formula for their number,…
Reinforcing a Philosophy: A counting approach to square functions over local fields
Kirsti D. Biggs, Julia Brandes, Kevin Hughes
In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of th…
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…
Rational lines on cubic hypersurfaces
Julia Brandes, Rainer Dietmann
We show that any smooth projective cubic hypersurface of dimension at least over the rationals contains a rational line. A variation of our methods provides a similar result o…
On the number of linear spaces on hypersurfaces with a prescribed discriminant
Julia Brandes
For a given form we apply the circle method in order to give an asymptotic estimate of the number of -tuples sp…
Vinogradov systems with a slice off
Julia Brandes, Trevor D. Wooley
Let denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\ldots +x_s^j=y_1^j+\ldots +y_s^j\quad (\text{, $j\ne…
The Hasse principle for systems of quadratic and cubic diagonal equations
Julia Brandes
Employing Brüdern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r_3 cubic and r_2 quadratic diagon…
Linear spaces on hypersurfaces over number fields
Julia Brandes
We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables requir…
Simultaneous additive equations: Repeated and differing degrees
Julia Brandes, Scott T. Parsell
We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulae, for systems of additive equations…
Forms representing forms: The definite case
Julia Brandes
Let and be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical repre…
Sums and differences of power-free numbers
Julia Brandes
We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions to the equations …
Integer points on homogeneous varieties with two or more degrees
Julia Brandes
We give a revised version of Schmidt's treatment of forms in many variables, which allows us to prove a Hasse principle under more lenient conditions on the number of variables tha…
Twins of s-free numbers
Julia Brandes
We generalise the square sieve developed by Heath-Brown to higher powers in order to improve on the error term for the problem of counting consecutive power-free numbers.
A note on p-adic solubility for forms in many variables
Julia Brandes
By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of no…
Forms representing forms and linear spaces on hypersurfaces
Julia Brandes
A generalisation of Waring's problem, considered first by Arkhipov and Karatsuba, is the question of representing not an integer, but a given polynomial, as a sum of powers of line…