Simultaneous Additive Equations: Repeated and Differing Degrees

Abstract

Abstract. 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 di ering degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning-HeathBrown, which give weak results when specialized to the diagonal situation, this is the rst result on such "hybrid" systems. We also obtain specialised results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.

Vinogradov Systems With a Slice Off

“…Both the aforementioned estimate I 4,3,1 (X ) X 4+ε , and the new bound reported in Theorem 1.1 go well beyond this work based on efficient congruencing and l 2 -decoupling. Indeed, when r = 1 we achieve an estimate tantamount to square-root cancellation in a range of 2s-th moments extending the interval 1 s k(k − 1)/2 roughly half way to the full conjectured range 1 s (k 2 + k − 2)/2. Our strategy for proving Theorem 1.1 is based on the proof of the estimate I 4,3,1 (X ) X 4+ε in [8,Theorem 1.3], though it is flexible enough to deliver estimates for the mean value I s,k,r (X ) with r 1, as we now outline.…”

“…In this paper we consider the number I s,k,r (X ) of integral solutions of the system of equations , and may be viewed as a testing ground for progress on systems not of Vinogradov type. Relatives of such systems have been employed in work on the existence of rational points on systems of diagonal hypersurfaces as well as cognate paucity problems (see for example [2][3][4]). The main conjecture for the system (1.1) asserts that whenever r, s, k ∈ N, r < k and ε > 0, then one may expect to be able to take ε to be zero.…”

The Hasse Principle for Systems of Quadratic and Cubic Diagonal Equations

“…It is commonly acknowledged that, unless fundamentally new ideas become available that avoid the implicit use of mean values, at least 6r 3 + 4r 2 + 1 variables are required in order to establish asymptotic estimates for the number of integral solutions of the system (1.1). This theoretical limit has recently been attained by Wooley [12, Theorem 1.1] in the case r 2 = r 3 = 1, and by the author jointly with Parsell [1,Theorem 1.4] for systems consisting of r 2 1 quadratic forms and one cubic equation. The latter work applies a disentangling argument going back in its essence to the methods of Davenport and Lewis [6], and provides estimates for the number of variables required to establish a Hasse principle and asymptotic formulae for the number of solutions of general systems of additive equations involving different degrees with arbitrary multiplicities.…”

“…For comparison, Theorem 1.4 of [1] establishes the same conclusion under the more stringent hypothesis that s 8r 3 + ⌊(8/3)r 2 ⌋ + 1, and proves a Hasse principle without asymptotic formula for s 7r 3 + ⌈(11/3)r 2 ⌉. Observe in particular that in the case r 2 = 1 Theorem 1.1 yields a bound on the number of variables given by s 6r 3 + 5 = 2(3r 3 + 2) + 1, so for systems of one quadratic and r 3 2 cubic equations we attain the theoretical limit imposed by square root cancellation.…”

“…S j (q, a) and singular integralJ(Y ) = [−Y P −2 ,Y P −2 ] r 2 [−Y P −3 ,Y P −3 ] r 3 s j=1 v j (β, P ) dβ.Then it follows from the arguments leading to equation (4.5) of[1] that N f (γ 1 ) · . .…”