What is the most general solution to an equation of the form: [tex]a_1 p_1 + \ldots + a_n p_n =0[/tex] where [tex]p_i[/tex] are given polynomials in several (N) variables with no common factor (i.e. their GCD is 1) and [tex]a_n[/tex] are the polynomials we are looking for (again in the same N variables). Of course, I'm asking for a nontrivial solution, i.e. not all a's are zero. For the case where n=2, i.e. where I only have [tex]a_1 p_1+a_2 p_2 =0[/tex], this is easy - we obtain that [tex]a_1=-c p_2, a_2=c p_1[/tex], where c is an arbitrary polynomial (recall that [tex]p_1, p_2[/tex] have no common factor). Does something simmilar hold also for n>2?