Cubic Diophantine inequalities for split forms. (English) Zbl 1372.11049
Let \(F(x)\) be a cubic form of \(s\) variables \(x:=(x_1,\dots,x_2)\), with real coefficients, and suppose that \(F\) splits into \(r\) non-zero cubic forms \(f_j\) \(1\leq j\leq r\), that is to say \(F(x)=\sum^r_{j=1}f_j(y_j)\) with \(f_j(y_j)\in\mathbb R[y_j]\backslash\{0\}\), \(y_1:=(y_{j1},\dots,y_{jn_j})\), \(y_{ji}\in\{x_k|1\leq k\leq s\}\), and \(y_{ji}\neq y_{kl}\) for \((j,i)\neq (k,l)\), following conditions are satisfied: 1) \(s\geq 358817445\); 2) \(s\geq 120897257\) and \(r=2\); 3) \(s\geq 35042291\) and \(r=3\); 4) \(s\geq 8324100\) and \(r=4\); 5) \(s\geq 120897257\) and \(r=5\); 6) \(s\geq 77027\) and \(r=6\). The author proves then that the form \(F(x)\) takes small values, that is \(| F(\alpha)|<1\) for some \(\alpha\) in \(\mathbb Z^a\backslash \{0\}\). These results are obtained by combining ideas developed in the works by D. E. Freeman [J. Lond. Math. Soc., II. Ser. 61, No. 1, 25–35 (2000; Zbl 0944.11011)] and by D. R. Heath-Brown [Proc. Lond. Math. Soc. (3) 100, No. 2, 560–584 (2010; Zbl 1233.11039)] with some numerical optimisation methods.
Reviewer: B. Z. Moroz (Bonn)
MSC:
| 11D75 | Diophantine inequalities |
| 11J25 | Diophantine inequalities |
| 11E76 | Forms of degree higher than two |
| 65K10 | Numerical optimization and variational techniques |
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