Simultaneous diagonal equations over p-adic fields

Brink, D.M.; Godinho, Hemar; Rodrigues, P.

doi:10.4064/aa132-4-8

Cited by 5 publications

(14 citation statements)

References 9 publications

(15 reference statements)

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“…Put

K = Q (a_{1}, \dots, a_{s})

. Then

K : Q

is a finite algebraic extension of

double-struckQ

, and it follows from [, Theorem 1], just as in the proof of [, Theorem 4.4], that in every completion

K_{v}

K

the equation has a non‐zero solution. Note that when the place

v

is infinite, this is trivially inferred from the hypothesis that, either

k

is odd, or else

L

is a totally imaginary extension of

double-struckQ

and

k

is even.…”

Section: Vinogradov's Mean Value Theorem In Number Fieldsmentioning

confidence: 99%

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

Wooley

2018

Proc. London Math. Soc.

122

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.

show abstract

“…Put

K = Q (a_{1}, \dots, a_{s})

. Then

K : Q

is a finite algebraic extension of

double-struckQ

, and it follows from [, Theorem 1], just as in the proof of [, Theorem 4.4], that in every completion

K_{v}

K

the equation has a non‐zero solution. Note that when the place

v

is infinite, this is trivially inferred from the hypothesis that, either

k

is odd, or else

L

is a totally imaginary extension of

double-struckQ

and

k

is even.…”

Section: Vinogradov's Mean Value Theorem In Number Fieldsmentioning

confidence: 99%

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

Wooley

2018

Proc. London Math. Soc.

122

141

View full text Add to dashboard Cite

show abstract

“…This first result on congruences modulo powers of

p

(especially Corollary 8), when combined with some of the analysis in 2, 5, 8 can be used to show that it suffices for

s

to be at least as large as some polynomial in both

d

and

R

(independent of

K

). The second result on congruences — and the results in the rest of the paper — are used to obtain a bound in Theorem A in which the exponent of both

R

and

d

is 2.…”

Section: Introductionmentioning

confidence: 95%

“…That a finite upper bound exists for the minimal number of variables required for a non‐trivial solution to a system () — at least for

R = 1

— was first proved by Birch 1. The bounds coming out of the methods of 1 were subsequently improved on in 2, 11, 12 and most recently by Moore in 9. In particular, the bounds proved in 9 show that it suffices to have

s > false(R d false) {(R m d + 1)}^{τ + 1}

p > 2

and

s > R d {(R m d + 1)}^{τ + 2}

p = 2

[9, Theorem 1.2].…”

Section: Introductionmentioning

confidence: 99%

“…To prove Theorem A in general we make use of the notions of equivalent and normalized systems of diagonal equations, much as introduced in the work of Davenport and Lewis 4 and exploited in Knapp 5 (and in 2) but somewhat tweaked to be more suited for our purposes. Using this, we show that any system such as () (if the associated invariant