Quadratic forms and systems of forms in many variables

Abstract

Let F 1 , . . . , F R be quadratic forms with integer coefficients in n variables. When n ≥ 9R and the variety V (F 1 , . . . , 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 , . . . , 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 d R. In the case that d ≥ 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. Mathematics Subject Classification

On the Hasse principle for complete intersections

“…There have been two recent notable breakthroughs. Myerson [16], [17] improved the square dependence on R in Birch's result to a linear one. When d = 2 and 3, these results improve the lower bound to n − σ ≥ 8R and 25R respectively.…”

Birch-Goldbach theorems

“…. , f r 都是二次形时, Rydin Myerson 问, 文献 [24] 中的证明方法能否用于改进 Cook 和 Magyar [37] 的结果. 如今自然可以问, 是否能够结合 Rydin Myerson 的方法和定理 5.5 的证明方法去 改进变量个数与 r 的关系.…”

Systems of cubic forms in many variables

“…To prove Theorem 1.2 we will use Theorem 1.3 from the author's previous work [17]. This will reduce the problem to proving an upper bound for the number of solutions to the following auxiliary inequality.…”

“…...,n ℓ=n−b+1,...,n . (5 17). In general(5.17) holds after permuting the rows and columns of the matrix H c (x) and one can then apply the same permutations throughout the rest of our construction of Y (i) , every time the matrix H c (x) appears.Define y (1) (x), .…”