Roots of multivariate polynomials

[Dragonfall]

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?

 

[micromass]

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.

 

[Dragonfall]

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.