INTEGERS REPRESENTED BY REVISITED

2020 ◽  
pp. 1-12
Author(s):  
MICHAEL A. BENNETT
Keyword(s):  
Quartic Form

We sharpen earlier work of Dabrowski on near-perfect power values of the quartic form $x^{4}-y^{4}$ , through appeal to Frey curves of various signatures and related techniques.

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

Determinantal representations of x4+y4+z4

1938 ◽  
Vol 34 (1) ◽  
pp. 6-21 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Theoretical Description ◽  
Fourth Degree ◽  
Linear Forms ◽  
Ternary Form ◽  
Quartic Form ◽  
Fourth Root

A ternary form of degree n can be expressed as a symmetrical determinant, of n rows and columns, whose elements are linear forms; furthermore, not only is such a mode of expression known to be possible, but A. C. Dixon, in 1902, gave* a process by which the determinant can be obtained when the ternary form is given. This process, however, although it admits of such a straightforward theoretical description, cannot be carried through in practice, for a general ternary form, without the introduction of complicated algebraical irrationalities, even if we restrict ourselves to forms of the fourth degree; consequently no application of Dixon's process to an actual example seems to have been published. If then a choice can be made of a quartic form for which the reduction to a symmetrical determinant can be carried out without undue complication, it seems fitting to give some account of it. The following pages are therefore devoted to the study, from this aspect, of the form x4+y4+z4, for which the reduction can be accomplished without introducing any irrationality other than the fourth root of − 1.

scholarly journals The minimum of a binary quartic form

1951 ◽  
Vol 84 (0) ◽  
pp. 263-298 ◽  
Author(s):  
C. S. Davis
Keyword(s):  
Quartic Form

scholarly journals The geometrical construction of Maschke's quartic surfaces

1945 ◽  
Vol 7 (2) ◽  
pp. 93-103 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Finite Order ◽  
Geometrical Construction ◽  
Remarkable Property ◽  
Quartic Surface ◽  
Quartic Form ◽  
Quartic Surfaces

There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.

scholarly journals An explicit estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a complete quartic form

2015 ◽  
Author(s):  
Yap Hong Keat ◽  
Kamel Ariffin Mohd Atan ◽  
Siti Hasana Sapar ◽  
Mohamad Rushdan Md Said
Keyword(s):  
Partial Derivative ◽  
Quartic Form ◽  
Common Zeros

scholarly journals A Novel Necessary and Sufficient Condition for the Positivity of a Binary Quartic Form

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Yang Guo
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Pencil Of Conics ◽  
Quartic Form ◽  
The Common ◽  
Necessary And Sufficient

In this paper, by considering the common points of two conics instead of the roots of the binary quartic form, we propose a novel necessary and sufficient condition for the positivity of a binary quartic form using the theory of the pencil of conics. First, we show the degenerate members of the pencil of conics according to the distinct natures of the common points of two base conics. Then, the inequalities about the parameters of the degenerate members are obtained according to the properties of the degenerate conics. Last, from the inequalities, we derive a novel criterion for determining the positivity of a binary quartic form without the discriminant.

NOTE ON A BINARY QUARTIC FORM

1950 ◽  
Vol 1 (1) ◽  
pp. 253-261 ◽  
Author(s):  
H. DAVENPORT
Keyword(s):  
Quartic Form

scholarly journals FERMION PRODUCTION IN STRONG MAGNETIC FIELD AND ITS ASTROPHYSICAL IMPLICATIONS

2007 ◽  
Vol 22 (25n28) ◽  
pp. 2081-2090 ◽  
Author(s):  
HYUN KYU LEE ◽  
YONGSUNG YOON
Keyword(s):  
Magnetic Field ◽  
Strong Magnetic Field ◽  
Imaginary Part ◽  
Critical Field ◽  
Effective Potential ◽  
Magnetic Field Gradient ◽  
Magnetic Moments ◽  
Field Configuration ◽  
Quartic Form ◽  
The Magnetic Field

We calculate the effective potential of a strong magnetic field induced by fermions with anomalous magnetic moments which couple to the electromagnetic field in the form of the Pauli interaction. For a uniform magnetic field, we find the explicit form of the effective potential. It is found that the non-vanishing imaginary part develops for a magnetic field stronger than a critical field and has a quartic form which is quite different from the exponential form of the Schwinger process. We also consider a linear magnetic field configuration as an example of inhomogeneous magnetic fields. We find that the imaginary part of the effective potential is nonzero even below the critical field and shows an exponentially decreasing behavior with respect to the inverse of the magnetic field gradient, which is the non-perturbative characteristics analogous to the Schwinger process. These results imply the instability of the strong magnetic field to produce fermion pairs as a purely magnetic effect. The possible applications to the astrophysical phenomena with strong magnetic field are also discussed.

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