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- W2319364974 abstract "where fi is a homogeneous cubic function of K, X, g. We thus secure a parametric representation of the points of the surface, which is therefore unicursal. In case all of the coefficients of the lij are rational, we have the complete solution in rational numbers of the Diophantine equation f = 0. However, there exist cubic equations f = 0 whose rational solutions involve three parameters homogeneously such that f is not expressible rationally in determinantal form (? 12). With the application to Diophantine analysis in mind, I here discuss algebraically the problem to express a given cubic form f as a determinant l ol Of the third order. Since we are interested ultimately in the case in which the coefficients of the lij are rational and since three linear functions of four variables vanish for values not all zero of the variables, we shall assume that f vanishes at a known rational point. Then the coefficients of the lij are shown to be expressible rationally in terms of a root of an algebraic equation whose leading coefficient is not zero if f = 0 has no singular point. The existence of a rational root may be decided by a finite number of trials. The corresponding binary problem is solved by the identity" @default.
- W2319364974 created "2016-06-24" @default.
- W2319364974 creator A5049229082 @default.
- W2319364974 date "1921-04-01" @default.
- W2319364974 modified "2023-09-25" @default.
- W2319364974 title "Algebraic Theory of the Expressibility of Cubic Forms as Determinants, with Application to Diophantine Analysis" @default.
- W2319364974 doi "https://doi.org/10.2307/2370244" @default.
- W2319364974 hasPublicationYear "1921" @default.
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