Thank you for your answer. When you multiply $y$ by $\left(1+\frac 16\sum_j u_j\right)^{-1/2}$, how would you have $\langle y,v_j\rangle\ge 1$ for all $j$? Thanks.

@gyashfe But this question is about the existence of solutions and the equation is generic. How could the algorithm that you referred to be used to get the correct answer to this question? Thanks.

@gyashfe Thank you for your comment. Could you please let us know what cylindrical algebraic decomposition is exactly and show us how should it possible to use it to get the correct answer to this question? Thanks.

@PraphullaKoushik Just wondering if you know the answer to the question or have any helpful idea that could lead an answer to the question. Thanks a lot.

The editing or rephrasing was due to no answer received for the original question. Best regards.

@PraphullaKoushik I tried the special cases of choosing some of them to be 0 's or 1's. I am not in the area of algebraic surface, but this question is about intersection of surfaces, which I believe the researchers in the area of surfaces or related can answer. Thanks.

@PraphullaKoushik I have tried special cases and the answer is yes for the special cases; and on the other hand, I could not find any counter-example. Best regards.