Cubic Forms as Sum of Cubes of Linear Forms Original Research Article

The topic of investigation is cubic formsFover image innvariables that are representable as a sumL31+L32of two cubes of linear forms with algebraic coefficients. IfZ2(n, X) denotes the number of such formsF, the main result, stated as Theorem 1.3, gives its order of magnitude asZ2(n, X)asymptotically equal toX2n/3, with the implied constants depending onnonly. The gist of our method consists of the analysis of thep-adic conditions for the coefficients of the linear formsL1andL2which stem from the fact thatFis defined over image. This leads to results concerning local lattices and their connection to global lattices that seem of interest even beyond the treated problem, and which are therefore stated with some more generality in Theorems 4.1 and 4.8. The combination of a fact from the Geometry of Numbers with the above then leads to the main theorem. Even though these steps are applied to changes of variables leading to the special diagonal formX3+Y3, they may be applicable to more general situations, the final goal being the treatment of forms that can be transformed into an arbitrary given formf(X, Y) by a suitable linear, algebraic change of variables. Another, probably difficult generalisation consists in increasing the number of variables to deal with formsX31+…+X3kforkgreater-or-equal, slanted3.