Cubic forms in thirty-two variables

1959 ◽  
Vol 251 (993) ◽  
pp. 193-232 ◽  
Keyword(s):  
Exponential Sum ◽  
Definite Integral ◽  
Cubic Form ◽  
New Form ◽  
Cubic Forms ◽  
Hardy Littlewood Method

It is proved that if C(xu...,*„) is any cubic form in n variables, with integral coefficients, then the equation C{xu ...,*„) = 0 has a solution in integers xXi...,xn, not all 0, provided n is at least 32. The proof is based on the Hardy-Littlewood method, involving the dissection into parts of a definite integral, but new principles are needed for estimating an exponential sum containing a general cubic form. The estimates obtained here are conditional on the form not splitting in a particular manner; when it does so split, the same treatment is applied to the new form, and ultimately the proof is made to depend on known results.

On a conjecture of Mordell concerning binary cubic forms

1941 ◽  
Vol 37 (4) ◽  
pp. 325-330 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Strict Inequality ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms

Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for whichThese inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent tofor the case D < 0, or tofor the case D > 0.

Cubic forms in 29 variables

1962 ◽  
Vol 266 (1326) ◽  
pp. 287-298 ◽  
Keyword(s):  
Cubic Form ◽  
Cubic Forms

It is proved that if C (x 1 ..., x n) is any cubic form in n variables, with integral coefficients, then the equation C ( x 1 x n ) = 0 has a solution in integers x 1 ..., x n not all 0, provided n is at least 29. This is an improvement on a previous result (Davenport 1959).

Cubic forms in sixteen variables

1963 ◽  
Vol 272 (1350) ◽  
pp. 285-303 ◽  
Keyword(s):  
Cubic Form ◽  
Cubic Forms

It is proved that if C ( x 1 , ..., x n ) is any cubic form in n variables, with integral coefficients, then the equation C ( x 1 , ..., x n ) = 0 has a solution in integers x 1 , ..., x n not all 0, provided n is at least 16. This is an improvement upon earlier results (Davenport 1959, 1962).

Slim Exceptional Sets for Sums of Cubes

2002 ◽  
Vol 54 (2) ◽  
pp. 417-448 ◽  
Author(s):  
Trevor D. Wooley
Keyword(s):  
Prime Numbers ◽  
Exponential Sum ◽  
Exceptional Sets ◽  
Hardy Littlewood Method

AbstractWe investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding X, that fail to have a representation as the sum of 7 cubes of prime numbers, is O(X23/36+ε). For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is O(X11/36+ε).

scholarly journals On the representations of numbers by binary cubic forms

1985 ◽  
Vol 27 ◽  
pp. 95-98 ◽  
Author(s):  
C. Hooley
Keyword(s):  
Positive Integer ◽  
Recent Work ◽  
Present Note ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms ◽  
Theory Of Numbers

There are not a few situations in the theory of numbers where it is desirable to have as sharp an estimate as possible for the number r(n) of representations of a positive integer n by an irreducible binary cubic formA variety of approaches are available for this problem but, as they stand, they are all defective in that they introduce unwanted factors into the estimate. For instance, an estimate involving the discriminant of f(x, y) is obtained if we adopt the Lagrange procedure [5] of using congruences of the type f(σ, 1)≡0, mod n, to reduce the problem to one where n=1. Alternatively, following Oppenheim (vid. [2]), Greaves [3], and others, we may appeal to the theory of factorization of ideals, which leads to unwanted logarithmic factors owing to the involvement of algebraic units. Having had need, however, in some recent work on quartic forms [4] for an estimate without such extraneous imperfections, we intend in the present note to prove thatuniformly with respect to the coefficients of f(x, y), where ds(n) denotes the number of ways of expressing n as a product of s factors.

scholarly journals Quadratic points of surfaces in projective 3-space

2019 ◽  
Vol 70 (3) ◽  
pp. 1105-1134
Author(s):  
Marcos Craizer ◽  
Ronaldo A Garcia
Keyword(s):  
Elliptic Surface ◽  
Line Field ◽  
Parabolic Curve ◽  
Cubic Form ◽  
Local Results ◽  
Cubic Forms ◽  
Quadratic Point

Abstract Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is ±1/3, while the index of the line field is ±1. Moreover, for an elliptic quadratic point whose cubic form is semi-homogeneous, we can use Loewner’s conjecture to show that the index is at most 1. From the above local results, we can conclude some global results: A generic compact elliptic surface has at least 6 quadratic points, a compact elliptic surface with semi-homogeneous cubic forms has at least 2 quadratic points and the number of quadratic points in a hyperbolic disc is odd. By studying the behavior of the cubic form in a neighborhood of the parabolic curve, we also obtain a relation between the indices of the quadratic points of a generic surface with non-empty elliptic and hyperbolic regions.

Cubic forms over algebraic number fields

1969 ◽  
Vol 66 (2) ◽  
pp. 323-333 ◽  
Author(s):  
C. Ryavec
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Rational Field ◽  
Cubic Form ◽  
Proper Subvariety ◽  
Almost All ◽  
Cubic Forms

In 1935 Tartakowski (7) proved that, in general, a cubic form in sufficiently many variables with coefficients in an algebraic number field K has a non-trivial zero in that field; and in the case when K is the rational field 57 variables suffice. Here, ‘in general’ means that the coefficients of the form do not lie in a proper subvariety of the coefficient space. Hence, Tartakowski's result holds for almost all cubic forms. Later, Lewis (5) proved that if K is any algebraic number field such that [K: Q] = n, then there exists a function ψ(n) such that every cubic form over K in m ≥ ψ(n) variables has a non-trivial zero in K. His bound, ψ(n), is extremely large; e.g. when K is the rational field, ψ(1) > 500.

scholarly journals On polynomials that equal binary cubic forms.

2006 ◽  
Vol Volume 29 ◽  
Author(s):  
C Hooley
Keyword(s):  
Cubic Form ◽  
A Value ◽  
International Audience ◽  
Cubic Polynomial ◽  
Cubic Forms ◽  
Rational Integral

International audience Let $F(x)$ be a cubic polynomial with rational integral coefficients with the property that, for all sufficiently large integers $n,\,F(n)$ is equal to a value assumed, through integers $u, v$, by a given irreducible binary cubic form $f(u,v)=au^3+bu^2v+cuv^2+dv^3$ with rational integral coefficients. We prove that then $F(x)=f(u(x),v(x))$, where $u=u(x), v=v(x)$ are linear binomials in $x$.

Cubic forms representing arithmetic progressions

1959 ◽  
Vol 55 (3) ◽  
pp. 270-273 ◽  
Author(s):  
G. L. Watson
Keyword(s):  
Arithmetic Progressions ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms

The following result has recently been proved by Lewis (3), Davenport (2) and Birch (1).There exists an integer n0 such that every cubic form with rational coefficients and at least n0 integral variables represents zero non-trivially.The arguments of Lewis and Birch are simple, and yield also various generalizations of this result. Davenport's proof is complicated, but it shows that the minimal n0 satisfies

XXIV. On the tangential of a cubic

1858 ◽  
Vol 148 ◽  
pp. 461-463
Keyword(s):  
Canonical Form ◽  
Second Line ◽  
Third Order ◽  
Cubic Form ◽  
The Third ◽  
Cubic Curve ◽  
Right Hand ◽  
The Right ◽  
Cubic Forms

In my “Memoir on Curves of the Third Order, ”I had occasion to consider a derivative which may be termed the "tangential” of a cubic, viz. the tangent at the point ( x , y , z ) of the cubic curve (*)( x , y , z ) 3 = 0 meets the curve in a point ( ξ , n , ζ ) which is the tangential of the first-mentioned point; and I showed that when the cubic is represented in the canonical form x 3 + y 3 + z 3 +6 lxyz = 0 , the coordinates of the tangential may be taken to be x ( y 3 — z 3 ) : y ( z 3 — x 3 ) : z ( x 3 — y 3 ). The method given for obtaining the tangen­tial may be applied to the general form ( a , b , c , f , i , j , k , l )( x , y , z ) 3 : it seems desirable, in reference to the theory of cubic forms, to give the expression of the tangential for the general form; and this is what I propose to do, merely indicating the steps of the calculation, which was performed for me by Mr. Creedy. The cubic form is ( a , b , c , f , g , h , i , j , k , l )( x , y , z ) 3 , which means ax 3 + by 3 + cz 3 +3 fy 2 z +3 gz 2 x +3 hx 2 y +3 iyz 2 +3 jzx 2 +3 hxy 2 +6 lxyz ; and the expression for ξ is obtained from the equation x 2 ξ =( b , f , i , c )( j , f , c , i , g , l )( x , y , z ) 2 ,—( h , b , i , f , l , k )( x , y , z ) 2 ) 3 — ( a , b , c , f , g , h , i , j , k , l )( x , y , z ) 3 (C x +D), where the second line is in fact equal to zero, on account of the first factor, which vanishes. And C, D, denote respectively quadric and cubic functions of ( y , z ), which are to be determined so as to make the right-hand side divisible by x 2 the resulting value of ξ may be modified by the adjunction of the evanescent term

Sign in / Sign up

Export Citation Format

Share Document