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 $l$

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Our results are generally comparable with recent advances in the field and supersede them in a number of cases.Comment: Withdrawn due to a crucial error on page 8.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 $F\in \mathbb Z[x_1,\dots,x_s]$ we apply the circle method in order to give an asymptotic estimate of the number of $m$-tuples $\mathbf x_1, \dots, \mathbf x_m$ spanning a linear space on the hypersurface $F(\mathbf x) = 0$ with the property that $\det ( (\mathbf x_1, \dots, \mathbf x_m)^t \, (\mathbf x_1, \dots, \mathbf x_m)) = b$. 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 $29$ 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 $u \ge 3v$ we obtain the Hasse principle for systems consisting of $v$ cubic and $u$ quadratic diagonal equations in $6v+4u+1$ variables, thus attaining the convexity barrier for this problem.Comment: Our original treatment of systems with degrees $k \ge 4$ 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 $k \ge 4$. All results involving only quadratic and cubic equations remain unaffecte

Vinogradov systems with a slice off

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\ldots +x_s^j=y_1^j+\ldots +y_s^j\quad (\text{$1\le j\le k$, $j\ne r$}),$$ with $1\le x_i,y_i\le X$ $(1\le i\le s)$. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\le r\le k-1$. In particular, when $s,k\in \mathbb N$ satisfy $k\ge 3$ and $1\le s\le (k^2-1)/2$, we establish the essentially diagonal behaviour $I_{s,k,1}(X)\ll X^{s+\epsilon}$.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