On Weyl's inequality and Waring's problem for cubes

Chowla, S.; Davenport, H

doi:10.4064/aa-6-4-505-521

Cited by 11 publications

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“…Moreover, Theorem 1 of Chowla and Davenport [10] establishes the same conclusion as Theorem 1 above in the special case where (x, y) is a binary cubic form with nonzero discriminant. In cases where d > 3, meanwhile, the conclusion of Theorem 1 provides substantially sharper estimates than would be available through the work of Birch [4] and Schmidt [25].…”

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confidence: 62%

“…Such arguments being rather lengthy, technical, and in any case limited in their application, we have chosen to present the main thrust of our ideas and defer any such considerations to a possible future occasion. We note that when d = 3 and the underlying form has nonzero discriminant, then one may establish the main conclusion of Theorem 2 through the methods of Chowla and Davenport [10] (see Lemma 4.1 of Brüdern and Wooley [9]). Finally, when d is larger than 11 or so, it is possible to apply a trivial argument involving Vinogradov's methods in order to obtain conclusions superior to those stemming from Theorem 2.…”

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confidence: 95%

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On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms

Wooley¹

1999

Duke Math. J.

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1. Introduction. The remarkable success enjoyed by the Hardy-Littlewood method in its application to diagonal diophantine problems rests in large part on the theory of exponential sums in a single variable. Following almost a century of intense investigations, the latter body of knowledge has reached a mature state that, although falling short of what is expected to be true, nonetheless suffices for the majority of applications. By contrast, exponential sums in many variables remain poorly understood, and, consequently, applications of the Hardy-Littlewood method to problems concerning the solubility of systems of forms in many variables are fraught with difficulties. While the methods of Weyl and of Vinogradov have been extended to estimate exponential sums in many variables (see, in particular, Arkhipov and Karatsuba [1], Arkhipov, Karatsuba and Chubarikov [2], Tartakovsky [26], Davenport [11], [12], [13], [14], Birch [4] and Schmidt [25]), in almost all circumstances the strength of the ensuing estimates is considerably inferior to that of corresponding estimates for Weyl sums in a single variable. Moreover, one is obliged to work under various hypotheses of a geometric nature. Indeed, it is only for exponential sums over polynomials diagonalisable over C and for nonsingular cubic polynomials and their close kin that we have estimates of strength to match those for corresponding exponential sums of a single variable (see Chowla and Davenport [10], Birch and Davenport [5], Heath-Brown [20] and Hooley [21], [22], [23]). The purpose of this paper is to develop estimates for exponential sums over binary forms of strength comparable to the best available for corresponding exponential sums of a single variable and, moreover, without any serious geometric hypotheses on the form. Since binary forms of degree exceeding 3 in general fail to diagonalise over C, it should be evident that our conclusions go beyond those of Birch and Davenport [5]. Our hope is that the methods described herein may spawn ideas for improved treatments of exponential sums in many variables.Before describing our main conclusions, we require some notation. Suppose that (x, y) ∈ Z[x, y] is a binary form of degree d exceeding 1. Then we say that is degenerate if there exist complex numbers α and β such that (x, y) is identically equal to (αx + βy) d . It is easily verified that when (x, y) is degenerate, then there

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confidence: 62%

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confidence: 95%

On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms

Wooley¹

1999

Duke Math. J.

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“…Chowla and Davenport [6] tried to generalise (1) by considering the form Hence no such solution can exist.…”

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confidence: 99%

“…Hence no such solution can exist. Chowla and Davenport [6] tried to generalise (1) by considering the form…”

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confidence: 99%

On certain cubic forms in seven variables

Harvey

2010

Math. Proc. Camb. Phil. Soc.

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“…The question now arises as to whether one can break away from the diagonal situation when the number of variables does not exceed 8. A first attempt was made by Chowla & Davenport (1961) over three decades ago. They considered binary cubic forms Φ j ∈ Z[x, y] (j = 1, 2, 3) with non-zero discriminant, and showed the existence of a non-trivial solution of the diophantine equation Φ 1 (x 1 , y 1 ) + Φ 2 (x 2 , y 2 ) + Φ 3 (x 3 , y 3 ) + ax 3 4 + by 3 4 = 0, where a, b ∈ Z.…”

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confidence: 99%

The addition of binary cubic forms

Brüdern

Wooley²

1998

Philosophical Transactions of the Royal Society of London. Seri

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We show that a sum of four non-degenerate binary cubic forms with integral coefficients necessarily possesses a non-trivial rational zero. When each of these binary cubic forms has non-zero discriminant, we are able to obtain bounds on the number, N (P ), of integral zeros of the sum inside a box of size P of the shape P 5−ε ε N (P ) ε P 5+ε . Finally, given two binary cubic forms with non-zero discriminant, we show that almost all integers, lying in those congruence classes permitted by local solubility conditions, are represented as the sum of the aforementioned forms.

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On Weyl's inequality and Waring's problem for cubes

Cited by 11 publications

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On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms

On Weyl’s inequality, Hua’s lemma, and exponential sums over binary forms

On certain cubic forms in seven variables

The addition of binary cubic forms

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