Polynomial equation in several variables

[GargleBlast42]

What is the most general solution to an equation of the form:

$$a_1 p_1 + \ldots + a_n p_n =0$$

where $$p_i$$ are given polynomials in several (N) variables with no common factor (i.e. their GCD is 1) and $$a_n$$ 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 $$a_1 p_1+a_2 p_2 =0$$, this is easy - we obtain that $$a_1=-c p_2, a_2=c p_1$$, where c is an arbitrary polynomial (recall that $$p_1, p_2$$ have no common factor). Does something simmilar hold also for n>2?

 

[chingkui]

There can be so many. If all you want is a nontrivial solution, then just use the a1, a2 as in your example, and set a3=...=an=0.

 

[GargleBlast42]

Well, that is certainly a solution, but I would like to obtain the most general form.

Could one, for example, show that all such solutions have to have the form $$a_i=\sum_{j\neq i} c_j p_j$$, with the $$c_j$$ being some polynomials obeying some further relation (which is obtained by substituting this ansatz to the equation).

I know probably to little from algebra to be able to prove something like that. Or maybe it's just trivial and I don't see it?