Common roots of multivariate polynomials

Click For Summary
SUMMARY

The discussion centers on efficiently solving the common roots of four polynomials in four variables. The user currently employs a gradient descent method, which is effective but raises concerns about local minima. They have explored the Cayley-Dixon and Macaulay resultant methods, but these require excessive memory. The conversation highlights Grobner bases and Buchberger's algorithm as promising alternatives for solving systems of polynomial equations.

PREREQUISITES
  • Understanding of polynomial equations and their roots
  • Familiarity with gradient descent optimization techniques
  • Knowledge of Cayley-Dixon and Macaulay resultant methods
  • Basic concepts of Grobner bases and Buchberger's algorithm
NEXT STEPS
  • Research Grobner bases and their applications in solving polynomial systems
  • Study Buchberger's algorithm for efficient computation of Grobner bases
  • Explore alternative optimization techniques to gradient descent for polynomial root finding
  • Investigate memory-efficient algorithms for polynomial resultants
USEFUL FOR

Mathematicians, computer scientists, and researchers focused on algebraic geometry, particularly those working on polynomial equations and optimization methods.

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.
 

Similar threads

Undergrad What can be deduced about the roots of this polynomial?
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Classification of reductive groups via root datum
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad If pair of polynomials have Greatest Common Factor as 1 ....
  • · Replies 1 ·
Replies
1
Views
2K
MHB Quartic polynomial has at least one real root
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Is quintic only polynomial that needs to be proven imposs?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Solution?: Quintic Equation from Physical System
  • · Replies 4 ·
Replies
4
Views
2K
MHB Finding roots of this particular polynomial
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Eigenvalues and characteristic polynomials
  • · Replies 10 ·
Replies
10
Views
3K
High School What are the applications of roots of a polynomial?
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Is \( u^p \) a root of \( f(x) \) over \( GF(p) \) if \( u \) is a root?
  • · Replies 3 ·
Replies
3
Views
3K