Roots of multivariate polynomials

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SUMMARY

The maximum number of roots of a multivariate polynomial over a field is infinite if the field is infinite, while it is limited to the number of elements in the field if it is finite. This discussion highlights the relationship between the degree of a polynomial and its roots, emphasizing that the question of whether a multivariate polynomial is identically zero is complex, particularly when considering finite fields. The challenge lies in determining if all coefficients of the polynomial are zero.

PREREQUISITES
  • Understanding of multivariate polynomials
  • Knowledge of finite and infinite fields
  • Familiarity with the fundamental theorem of algebra
  • Concept of polynomial coefficients
NEXT STEPS
  • Research the properties of multivariate polynomials
  • Explore the implications of the fundamental theorem of algebra in multivariate contexts
  • Study the characteristics of finite fields in relation to polynomial roots
  • Investigate algorithms for determining if a polynomial is identically zero
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Mathematicians, computer scientists, and researchers in algebraic geometry who are exploring the properties of multivariate polynomials and their roots.

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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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.
 
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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