146 research outputs found
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 to the equations and , where is -free and is
-free. This is the first time that this problem has been studied with
distinct powers and
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and supersede them in a number of cases.Comment: Withdrawn due to a crucial error on page 8. Thanks to D.R.
Heath-Brown for spotting thi
On the number of linear spaces on hypersurfaces with a prescribed discriminant
For a given form we apply the circle method
in order to give an asymptotic estimate of the number of -tuples spanning a linear space on the hypersurface with the property that . This allows us in some measure to
count rational linear spaces on hypersurfaces whose underlying integer lattice
is primitive
Rational lines on cubic hypersurfaces
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 over p-adic fields. In both cases, we improve on
previous results due to the second author and Wooley.
We include an appendix in which we highlight some slight modifications to a
recent result of Papanikolopoulos and Siksek. It follows that the set of
rational points on smooth projective cubic hypersurfaces of dimension at least
29 is generated via secant and tangent constructions from just a single point.Comment: An oversight in Lemma 3.1 as well as a few typos have been correcte
Optimal mean value estimates beyond Vinogradov's mean value theorem
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 conjectured conclusions. This is the first occasion on
which bounds of this quality have been attained for Diophantine systems not of
Vinogradov type. As a consequence of this progress, whenever we
obtain the Hasse principle for systems consisting of cubic and
quadratic diagonal equations in variables, thus attaining the
convexity barrier for this problem.Comment: Our original treatment of systems with degrees contained a
fatal flaw (thanks to S. T. Parsell for alerting us to this). The revised
version gives an adapted treatment, leading to different results for . All results involving only quadratic and cubic equations remain
unaffecte
Vinogradov systems with a slice off
Let denote the number of integral solutions of the modified
Vinogradov system of equations with .
By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for
for . In particular, when
satisfy and , we establish the essentially
diagonal behaviour .Comment: 19 page
IMF's assistance: Devil's kiss or guardian angel?
This paper contributes to the debate on the efficacy of IMF's catalytic finance in preventing financial crises. Extending Morris and Shin (2006), we consider that the IMF's intervention policy usually exerts a signaling effect on private creditors and that several interventions in sequence may be necessary to avert an impending crisis. Absent of the IMF's signaling ability, our results state that repeated intervention is required to bail out a country, where by additional assistance may induce moral hazard on the debtor side. Contrarily, if the IMF exerts a strong signaling effect, one single intervention suffices to avoid liquidity crises. --catalytic finance,debtor moral hazard,global games
- …