I have a finite field $F_p$ and $m$ polynomials $P_1, P_2, \dots, P_m$ in this field. Also every polynomial is multivariable, so $P_i$ from $F_p^n$ to $F_p.$ It is known that $n>\sum_i \deg(P_i).$ And in the point $(c_1, c_2, \dots, c_n)$ every polynomial is zero. I need prove that it is not only single point in which every polynomial is zero.
I think there can be useful variable method of proving. It is method in which if in some set any object has property A with probability >0, then here exist A-object.