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Today, I came to know about homotopy continuation method to solve system of multivariate polynomials. This method finds its roots from the field of Numerical algebraic geometry.

I already know that Groebner bases methods are also used to solve system of multivariate polynomials.

I want to know which method is faster. Is there some kind of complexity analysis available in literature. What are pros and cons of these? Are there cases where one method is superior over the other?

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    $\begingroup$ The answer depends a lot on the specific question: do you want one solution or all? Is the variety zero dimensional? How do you deal with numerical instability, what is the computational model (complex numbers cannot be stored on an ordinary computer) etc. $\endgroup$
    – Thomas Kahle
    Commented Sep 26, 2021 at 17:07

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