Solving polynomial systems is known to be a NP hard problem; however it is not completely clear to me where this complexity comes from.

Intuitively, I can imagine that part of the problem consists in the counting of the roots of the system: Bezout bound states that the number of roots is bounded by the product of the degrees of the polynomials composing the system; so if we have $n$ polynomials of degree 2, we will have at most $2^n $ solutions, so, if $t$ is the time needed to compute a solution, we should perform at most

$$ t2^n $$

operations. My question is:

1. What can be said about t , the complexity of calculating a single solution?
2. Is still a NP hard problem to solve non linear polynomial systems that are known to have only one solution?
3. Is there any literature about this topic, studying the complexity of solving polynomial systems?

thanks!

Solving polynomial systems is known to be a NP hard problem; however it is not completely clear to me where this complexity comes from.

My interest is in the case of systems of multivariate polynomials over the real field.

Intuitively, I can imagine that part of the problem consists in the counting of the roots of the system: Bezout bound states that the number of roots is bounded by the product of the degrees of the polynomials composing the system; so if we have $n$ polynomials of degree 2, we will have at most $2^n $ solutions, so, if $t$ is the time needed to compute a solution, we should perform at most

$$ t2^n $$

operations. My question is:

1. What can be said about t , the complexity of calculating a single solution?
2. Is still a NP hard problem to solve non linear polynomial systems that are known to have only one solution?
3. Is there any literature about this topic, studying the complexity of solving polynomial systems?

thanks!