Rational points on cubic hypersurfaces that split into four forms (Q2810733)

This paper studies the existence of a non-trivial rational point on the cubic hypersurface NEWLINE\[NEWLINEX: C(x_1,\ldots, x_n)=0,NEWLINE\]NEWLINE where \(C(x_1,\ldots, x_n)\) is a cubic form with integer coefficients. It is known that there are non-trivial solutions in every completion \(\mathbb{Q}_p\) if \(n\geq 10\), and hence if one assumes that the Hasse principle holds in this range, one would expect that \(X(\mathbb{Q})\neq \emptyset\) for \(n\geq 10\) (viewing \(X\) as a projective variety in \(\mathbb{P}^{n-1}\)). It was proved by \textit{D. R. Heath-Brown} [Proc. Lond. Math. Soc. (3) 47, 225--257 (1983; Zbl 0494.10012)] that \(X\) has indeed a rational point if \(n\geq 10\) and if \(C\) is a non-singular cubic form. Without any restrictions on \(C\), [\textit{D. R. Heath-Brown}, Invent. Math. 170, No. 1, 199--230 (2007; Zbl 1135.11031)] established the existence of a non-trivial rational solution if \(n\geq 14\), in combining geometric arguments with van der Corput differencing and the circle method.NEWLINENEWLINEThe circle method can already be applied in viewer variables if one assumes that the form \(C(x_1,\ldots, x_n)\) splits off a number of variables. If \(C\) can be written as \(C_1(x_1,\ldots, x_m)+C_2(x_{m+1},\ldots, x_n)\) with \( 1\leq m < n\) and cubic forms \(C_1\) and \(C_2\), then \textit{T. D. Browning} and \textit{J. L. Colliot-Thélène} [Compos. Math. 146, No. 4, 853--885 (2010; Zbl 1198.14021)] showed that \(X(\mathbb{Q})\neq \emptyset\) as soon as \(n\geq 13\). This idea was further generalized in work of \textit{B. Xue} and \textit{H. Dai} [Bull. Lond. Math. Soc. 46, No. 1, 169--184 (2014; Zbl 1301.11039)] to cubic hypersurfaces that split into three non-trivial cubic forms and \(n\geq 11\). More generally one says that \(C\) splits into \(r\) cubic forms, if \(C\) can be written (possibly after a linear transformation of the variables) in the form NEWLINE\[NEWLINEC(x_1,\ldots, x_n)= C_1+\ldots + C_r,NEWLINE\]NEWLINE where \(C_1,\ldots, C_r\) are cubic forms that have a positive number of independent variables. The paper under review refines the earlier results on cubic hypersurfaces splitting of forms for the case \(r=4\). The main theorem establishes \(X(\mathbb{Q})\neq \emptyset\) as soon as \(C\) splits into four forms and \(n\geq 10\).NEWLINENEWLINEThe proof builds on earlier developments on cubic hypersurfaces splitting off forms as mentioned above. In particular, it follows from [loc. cit.] that the only case that is left to consider is where \(C\) has the shape NEWLINE\[NEWLINEC(x_1,\ldots, x_{10})=c_1x_1^3+c_2x_2^3+c_3x_3^3+C_0(x_4,\ldots, x_{10}),NEWLINE\]NEWLINE with \(C_0\) an integral cubic form in \(7\) variables and \(c_1,c_2,c_3\in \mathbb{Z}\setminus\{0\}\). Furthermore, one may assume that \(C_0\) is `good' in the sense that NEWLINE\[NEWLINE\sharp\{\mathbf{x}\in \mathbb{Z}^7: |\mathbf{x}|\leq B: \text{rank}H(\mathbf{x})=r\}\ll B^{r+\epsilon},NEWLINE\]NEWLINE where \(H(\mathbf{x})\) is the Hessian of \(C_0\) (as otherwise \(C=0\) would have non-trivial solutions for geometric reasons).NEWLINENEWLINEInteger solutions to \(C=0\) are now detected via the Hardy-Littlewood circle method. As the authors only aim for lower bounds on the number of solutions in a box, one may restrict some of the variables to suitable subsets of the integers. Here, the authors restrict the first two variables \(x_1\) and \(x_2\) to \(\eta\)-smooth numbers for a suitable parameter \(\eta\). This has the advantage that one can use a strong \(6\)th moment on cubic Weyl sums of \textit{T. D. Wooley} [Mathematika 47, No. 1--2, 53--61 (2000; Zbl 1026.11075)]. The proof then proceeds using a form of pruning to different sized major/minor arcs. In a box of size \(|\mathbf{x}|\leq P\) one finally finds at least \(\gg P^7\) solutions to \(C(x_1,\ldots, x_{10})=0\).