Rational solutions of pairs of diagonal equations, one cubic and one quadratic

Wooley, Trevor D.

doi:10.1112/plms/pdu054

Cited by 26 publications

(32 citation statements)

References 19 publications

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“…Our starting point for the proof of case (ii) of the theorem is again the upper bound . By appealing to a standard transference principle (see [, Lemma 14.1]), one deduces that whenever

b \in Z

and

r \in N

satisfy

(b, r) = 1

and

false| α_{1} - b / r false| ⩽ r^{- 2}

, then one has

Ψ false(α_{1}, β_{κ - 1} false) ≪ X^{κ - 1 + ε} false(λ^{- 1} + X^{- 1} + λ X^{- κ} false),

where

λ = r + X^{κ} false| r α_{1} - b false|

. When

α \in {frakturM}_{κ} false(r, b false) \subseteq {frakturM}_{κ}

, moreover, one has both

r ⩽ X

and

X^{κ} false| r α_{1} - b false| ⩽ X

, so that

λ ⩽ 2 X

.…”

Section: Analogues Of Hua's Lemma and Waring's Problemmentioning

confidence: 99%

Nested efficient congruencing and relatives of Vinogradov's mean value theorem

Wooley

2018

Proc. London Math. Soc.

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121

141

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We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when φj∈Zfalse[tfalse] (1⩽j⩽k) is a system of polynomials with non‐vanishing Wronskian, and s⩽k(k+1)/2, then for all complex sequences (frakturan), and for each ε>0, one has 0true∫[0,1)k0true∑|n|⩽Xfrakturanefalse(α1φ1(n)+⋯+αkφk(n)false)2sdα≪Xε0true∑|n|⩽X|frakturan|2s.As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents k, recovering the recent conclusions of the author (for k=3) and Bourgain, Demeter and Guth (for k⩾4). In contrast with the l2‐decoupling method of the latter authors, we make no use of multilinear Kakeya estimates, and thus our methods are of sufficient flexibility to be applicable in algebraic number fields, and in function fields. We outline such extensions.

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“…Our starting point for the proof of case (ii) of the theorem is again the upper bound . By appealing to a standard transference principle (see [, Lemma 14.1]), one deduces that whenever

b \in Z

and

r \in N

satisfy

(b, r) = 1

and

false| α_{1} - b / r false| ⩽ r^{- 2}

, then one has

Ψ false(α_{1}, β_{κ - 1} false) ≪ X^{κ - 1 + ε} false(λ^{- 1} + X^{- 1} + λ X^{- κ} false),

where

λ = r + X^{κ} false| r α_{1} - b false|

. When

α \in {frakturM}_{κ} false(r, b false) \subseteq {frakturM}_{κ}

, moreover, one has both

r ⩽ X

and

X^{κ} false| r α_{1} - b false| ⩽ X

, so that

λ ⩽ 2 X

.…”

Section: Analogues Of Hua's Lemma and Waring's Problemmentioning

confidence: 99%

Nested efficient congruencing and relatives of Vinogradov's mean value theorem

Wooley

2018

Proc. London Math. Soc.

Self Cite

121

141

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“…where ρ denotes Dickman's function. We recall from the arguments of [19, Theorem 7.3] and [21,Lemma 8.6] (see also [14], It then follows easily that when α = a/q + β ∈ N(q, a, X) ⊆ N(X), one has [−XP −k l,n , XP −k l,n ] µ l .…”

Section: The Major Arcsmentioning

confidence: 99%

“…When the degrees are the same, the classical approach is to make a linear change of variables so that the mean values factor into a product of one-dimensional integrals, as in the work of Davenport and Lewis [12], Cook [10], [11], and Brüdern and Cook [5], though recently new ideas have become available in the work of Brüdern and Wooley [6,7,9,8]. Meanwhile, when the d j are distinct, such investigations are made possible by the iterative method of Wooley [21], [22], [23], which yields mean value estimates for exponential sums of the shape f k (α; A) = x∈A e(α 1 x k 1 + · · · + α t x kt ) when A is a set of suitably smooth integers. Here the second author [15] has obtained bounds for pairs of equations in a handful of particular cases by optimizing over a large collection of iterative schemes in the style of Vaughan and Wooley [20].…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Additive Equations: Repeated and Differing Degrees

Brandes

Parsell

2017

Can. j. math.

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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.

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“…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. For each integral solution x, y of the system (1.1) with 1 x, y X , one has the additional equation…”

Section: In Memoriam Klaus Friedrich Rothmentioning

confidence: 99%