arXiv subjects

Julia Brandes

Publications and source records attributed to Julia Brandes.

24 records · Page 2Linked to original sources

arXiv2015

Simultaneous additive equations: Repeated and differing degrees

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 containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning--Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems.

math.NT↗

arXiv2015

Sums and differences of power-free numbers

We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions $a, b \in \mathbb N$ to the equations $a+b=n$ and $a-b=n$, where $a$ is $k$-free and $b$ is $l$-free. This is the first time that this problem has been studied with distinct powers $k$ and $l$.

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arXiv2015

A note on p-adic solubility for forms in many variables

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 non-trivial p-adic solutions suffices to establish positivity of the singular series. This improves on earlier approaches due to Davenport, Schmidt and others, and allows us to establish an asymptotic formula for the number of simultaneous zeros of non-singular pairs of cubic forms in at least 131 variables. As a by-product, we obtain a version of Hensel's Lemma for linear spaces.

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arXiv2014

Integer points on homogeneous varieties with two or more degrees

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 than what had previously been thought possible with these methods. Our results are generally comparable with recent advances in the field and supersede them in a number of cases.

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arXiv2013

Forms representing forms and linear spaces on hypersurfaces

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 linear polynomials. We investigate a related problem and prove a Hasse principle for the number of identical representations of a set of given forms by homogeneous polynomials of general shape. The result leads to sizeable improvements for estimates of the number of linear spaces on the intersection of hypersurfaces.

math.NT↗

arXiv2013

Twins of s-free numbers

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.

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