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Infinity. For example, take (almost) any multivariate polynomial. That is if the field is infinite.What is the maximum number of roots of a multivariate polynomial over a field?

If the field is finite, then it can have at most the number of elements in the field.

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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.

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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