Polynomial equation in several variables

GargleBlast42
Messages
28
Reaction score
0
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?
 
Physics news on Phys.org
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.
 
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?
 

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Similar threads

MHB What are the properties of affine bases in $\mathbb{R}^n$?
Replies
15
Views
2K
I If L is diagonalizable, algebraic & geometric multiplicities are equal
Replies
4
Views
3K
I 1D time dependent PDE - using orthogonal collocation
Replies
2
Views
3K
A Solutions of Matrix Equation A X B^T = C
Replies
1
Views
2K
MHB Gcd of polynomials is 1. There is an nxn matrix with determinant....
Replies
5
Views
2K
MHB Prove that there are infinitely many irreducible polynomials in Z5.
Replies
3
Views
3K
A Is the proof of these results correct?
Replies
28
Views
3K
Constructing cubic spline interpolation polynomials
Replies
4
Views
1K
I Confused about small detail in rank-nullity theorem
Replies
1
Views
1K
MHB Polynomial in n variables: Prove the identity
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top