Products of consecutive values of some quartic polynomials

2019 ◽  
Vol 60 (2) ◽  
pp. 237-244
Author(s):  
Artūras Dubickas
Keyword(s):  
Quartic Polynomials

scholarly journals Irreducible Quartic Polynomials with Factorizations modulo p

2005 ◽  
Vol 112 (10) ◽  
pp. 876 ◽  
Author(s):  
Eric Driver ◽  
Philip A. Leonard ◽  
Kenneth S. Williams
Keyword(s):  
Modulo P ◽  
Quartic Polynomials

XI—The Identification of Klein's Quartic

1944 ◽  
Vol 62 (1) ◽  
pp. 83-91
Author(s):  
W. L. Edge
Keyword(s):  
Standard Form ◽  
Geometrical Interpretation ◽  
The Other ◽  
Veronese Surface ◽  
Quartic Polynomial ◽  
New Form ◽  
Quartic Curves ◽  
Quartic Polynomials

SummaryThere is a mode of specialising a quartic polynomial which causes a binary quartic to become equianharmonic and a ternary quartic to become a Klein quartic, admitting a group of 168 linear self-transformations. The six relations which must be satisfied by the coefficients of the ternary quartic were given by Coble forty years ago, but their true significance was never suspected and they have remained until now an isolated curiosity. In § 2 we give, in terms of a quadric and a Veronese surface, the geometrical interpretation of the six relations; we also give, in terms of the adjugate of a certain matrix, their algebraical interpretation. Both these interpretations make it abundantly clear that this set of relations specialising a ternary quartic has analogues for quartic polynomials in any number of variables, and point unmistakably to what these analogues are.That a ternary quartic is, when so specialised, a Klein quartic is proved in §§ 4–6. The proof bifurcates after (5.3); one branch leads instantly to the standard form of the Klein quartic while the other leads to another form which, on applying a known test, is found also to represent a Klein quartic. One or two properties of the curve follow from this new form of its equation. In §§ 8–10 some properties of a Veronese surface are established which are related to known properties of plane quartic curves; and these considerations lead to a discussion, in § 11, of certain hexads of points associated with a Klein curve.

Displacement Analysis of Two Special Cases of the TTTR Mechanisms

1989 ◽  
Vol 111 (3) ◽  
pp. 309-314 ◽  
Author(s):  
Wei Lin ◽  
Joseph Duffy
Keyword(s):  
Angular Displacement ◽  
Displacement Analysis ◽  
Special Cases ◽  
Quartic Polynomials

A pair of quartic polynomials in the tangent of the output angular displacement were derived for two TTTR mechanisms designed in such a way that the serially connected pairs of Hooke joints have their successive transverse axes parallel or mutually perpendicular. These results form the basis for an efficient reverse displacement analysis and for the design of a new group of robots with three serially connected Hooke joints which may prove to be attractive to industry.

scholarly journals Irreducible Quartic Polynomials with Factorizations modulop

2005 ◽  
Vol 112 (10) ◽  
pp. 876-890
Author(s):  
Eric Driver ◽  
Philip A. Leonard ◽  
Kenneth S. Williams
Keyword(s):  
Quartic Polynomials
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