Roots of multivariate polynomials

In summary, the maximum number of roots of a multivariate polynomial over a field is infinity if the field is infinite and can have at most the number of elements in the field if the field is finite. There is a relationship between degree and number of roots. However, determining whether a multivariate polynomial is identically zero can be a hard problem, especially when the polynomial is over a finite field and all coefficients must be 0.
  • #1
Dragonfall
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What is the maximum number of roots of a multivariate polynomial over a field? Is there a multivariate version of the fundamental theorem of algebra?
 
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  • #2
Dragonfall said:
What is the maximum number of roots of a multivariate polynomial over a field?

Infinity. For example, take (almost) any multivariate polynomial. That is if the field is infinite.

If the field is finite, then it can have at most the number of elements in the field.
 
  • #3
I said the maximum number of roots of a polynomial, not the maximum of all polynomials.

In other words is there a relationship between degree and number of roots.

But the question isn't relevant now. I was wondering why determining whether a multivariate polynomial is identically zero a hard problem. Turns out the problem actually wants to know whether all coefficients are 0 and the polynomial is over a finite field.
 
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What are the roots of a multivariate polynomial?

The roots of a multivariate polynomial are the values of the variables that make the polynomial equal to zero. For example, in the polynomial 3x^2 + 2xy + y^2, the roots would be x = 0 and y = 0.

How do you find the roots of a multivariate polynomial?

There is no general formula for finding the roots of a multivariate polynomial. However, there are various techniques that can be used, such as factoring, substitution, and numerical methods.

Can a multivariate polynomial have complex roots?

Yes, a multivariate polynomial can have complex roots. Just like in single-variable polynomials, the roots of a multivariate polynomial can be real or complex numbers.

What is the degree of a multivariate polynomial?

The degree of a multivariate polynomial is the highest sum of the exponents for each term in the polynomial. For example, in the polynomial 3x^2 + 2xy + y^2, the degree would be 3 (2+1).

What are the applications of studying the roots of multivariate polynomials?

Studying the roots of multivariate polynomials has many practical applications, such as in optimization problems, curve fitting, and solving systems of equations. It is also used in fields like engineering, economics, and physics.

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