Least zero of pairs of additive cubic equations

Abstract An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms (1) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mi>F</m:mi> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>n</m:mi> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mi>G</m:mi> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>n</m:mi> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \displaystyle{}\left\{\begin{aligned} \displaystyle{}F&amp;\displaystyle=c_{1}x_{1% }^{3}+c_{2}x_{2}^{3}+\cdots+c_{n}x_{n}^{3}=0,\\ \displaystyle G&amp;\displaystyle=d_{1}x_{1}^{3}+d_{2}x_{2}^{3}+\cdots+d_{n}x_{n}^% {3}=0,\\ \end{aligned}\right. under the “ M -good” condition for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>16</m:mn> </m:mrow> </m:math> {n\geq 16} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>c</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> {c_{1},\dots,c_{n}} and <m:math xmlns:m="http://www.w3.org/1