Quadratic forms and systems of forms in many variables

Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for $V(F_1,\dotsc,F_R)$. Previous work in this direction required $n$ to grow at least quadratically with $R$. We give a similar result for $R$ forms of degree $d$, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as $n> d2^dR$. In the case that $d\geq 3$, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients.Comment: 29 pages, in revie

Systems of cubic forms in many variables

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular we can handle systems of forms in $O(R)$ variables, previous work having required that $n \gg R^2$. One conjectures that $n \geq 6R+1$ should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.Comment: 23 pages, submitte

Bounds for spectral projectors on tori

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of Lp norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it

Strichartz estimates for the Schroedinger equation on non-rectangular two-dimensional tori

We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) at tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems

Repulsion : a how-to guide

Consider the integral zeroes of one or more, not necessarily diagonal, integral polynomials in many variables with the same degree. The basic principles for applying the circle method here were laid out by Birch. One way to improve on his work is repulsion: showing that the exponential sum over the polynomials can be large only on small, well separated regions. Unusually for improvements on Birch’s work this idea has been successfully applied to systems which are not particularly singular and which contain many polynomials. To begin I will ask: what about Birch’s work suggests that repulsion could be an improvement? I will then discuss the quadratic and higher degree case in detail, and an application to systems of forms with real coefficients

Systems of forms in many variables

We consider systems of polynomial equations and inequalities to be solved in integers. By applying the circle method, when the number of variables is large and the system is geometrically well-behaved we give an asymptotic estimate for the number of solutions of bounded size. In the case of R homogeneous equations having the same degree d, a classic theorem of Birch provides such an estimate provided the number of variables is R(R+1)(d-1)2d-1+R or greater and the system is nonsingular. In many cases this conclusion has been improved, but except in the case of diagonal equations the number of variables needed has always grown quadratically in R. We give a result requiring only d2dR+R variables, obtaining linear growth in R. When d = 2 or 3 we require only that the system be nonsingular; when d&amp;LT;4 we require that the coefficients of the equations belong to a certain explicit Zariski open set. These conditions are satisfied for typical systems of equations, and can in principle be checked algorithmically for any particular system. We also give an asymptotic estimate for the number of solutions to R polynomial inequalities of degree d with real coefficients, in the same number of variables and satisfying the same geometric conditions as in our work on equations. Previously one needed the number of variables to grow super-exponentially in the degree d in order to show that a nontrivial solution exists.</p

Bounds for spectral projectors on generic tori

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on generic tori, including generic rectangular tori. We state a conjecture and partially prove it, improving on previous results concerning arbitrary tori

Bounds for spectral projectors on generic tori

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on generic tori, including generic rectangular tori. We state a conjecture and partially prove it, improving on previous results concerning arbitrary tori.Comment: 23 page