Common roots of multivariate polynomials

In summary, the conversation is about efficiently solving the common root of 4 polynomials in 4 variables algebraically. The speaker is currently using a gradient descent method, but is concerned about local minima. They have also tried the Caylay-Dixon and Macaualy resultant methods, but they require too much memory. The other person suggests learning about Grobner bases and Buchberger's algorithm, which can be used for solving systems of polynomial equations.
  • #1
psyloe
1
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?
 
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  • #2
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.
 

FAQ: Common roots of multivariate polynomials

What are common roots of multivariate polynomials?

Common roots of multivariate polynomials refer to the values or numbers that satisfy all of the given polynomials when substituted for the variables. In other words, these are the values that make all of the polynomials equal to zero.

How are common roots of multivariate polynomials found?

Common roots of multivariate polynomials can be found by using methods such as the substitution method, the elimination method, or the graphical method. These methods involve solving the polynomials simultaneously to find the values that make them equal to zero.

Why are common roots of multivariate polynomials important?

Common roots of multivariate polynomials are important because they can help in solving systems of equations, which have many real-life applications. They can also help in simplifying complex polynomials and finding the factors of a polynomial.

Can there be multiple common roots of multivariate polynomials?

Yes, there can be multiple common roots of multivariate polynomials. In fact, the number of common roots can be equal to the degree of the polynomials. For example, if two polynomials of degree 3 have three common roots, then they have all their roots in common.

How do common roots of multivariate polynomials relate to the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra states that a polynomial of degree n has n complex roots. This includes multiple roots or common roots. Therefore, the number of common roots of multivariate polynomials can be determined by their degree, which is related to the Fundamental Theorem of Algebra.

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