Common roots of multivariate polynomials

psyloe
Messages
1
Reaction score
0
I was wondering if it were possible to efficiently solve the common root of 4 polynomials in 4 variables algebraically. I am currently using a gradient descent method, which can find these roots in a couple seconds; however, I am concerned about local minima.

So far I have attempted to use the Caylay-Dixon and Macaualy resultant to solve this problem, but these methods take far more memory to compute than is available. Is there a method that is more efficient than the ones I have tried?
 
Physics news on Phys.org
I think that learning about Grobner bases / Buchberger's algorithm will help you.

There are algorithms based on these concepts for solving systems of polynomial equations.
 

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 »

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 '##(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 »

Similar threads

I What can be deduced about the roots of this polynomial?
Replies
3
Views
3K
MHB Quartic polynomial has at least one real root
Replies
2
Views
3K
I Is quintic only polynomial that needs to be proven imposs?
Replies
2
Views
2K
A Solution?: Quintic Equation from Physical System
Replies
4
Views
2K
Eigenvalues and characteristic polynomials
Replies
10
Views
3K
MHB Finding roots of this particular polynomial
Replies
4
Views
2K
B What are the applications of roots of a polynomial?
Replies
5
Views
3K
Is \( u^p \) a root of \( f(x) \) over \( GF(p) \) if \( u \) is a root?
Replies
3
Views
3K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
Replies
7
Views
2K
Roots of polynomials as nonlinear systems of equations
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top