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Polynomial equation in several variables

  1. Jun 9, 2010 #1
    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?
  2. jcsd
  3. Jun 9, 2010 #2
    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.
  4. Jun 9, 2010 #3
    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 [tex]a_i=\sum_{j\neq i} c_j p_j[/tex], with the [tex]c_j[/tex] 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?
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