Given a system of complex polynomial equations, we seek the solution set. If we have more equations than variables, then we might expect a finite solution set. One may obtain the solution set by computing a Groebner basis, but this is slow in practice. Alternatively, one may leverage ideas from numerical algebraic geometry to obtain numerical solutions. Depending on the system, one might then apply LLL-type algorithms to promote these numerical solutions to exact solutions. But can you tell whether you have the full solution set?
Question: Given finite $F\subseteq\mathbb{C}[X_1,\ldots,X_n]$ and a finite set $S\subseteq\mathbb{C}^n$ of common zeros of $F$, is there an efficient way to certify whether $S$ contains every common zero of $F$?
The case of a single monic polynomial in one variable is straightforward: Take derivatives to deduce the multiplicity of each root, and then multiply binomial factors to obtain the original polynomial. This certifies completeness of the solution set.
When there are more variables, things are less obvious. In some cases, Bezout's theorem gives the size of the solution set (counted with multiplicity) provided the solution set is finite. Presumably, one may use calculus to determine the multiplicity of a given solution, but it's unclear how to rule out an infinite solution set.
To be clear, I am only interested in computational approaches that are (much) faster than Groebner basis calculation, at least in practice.
Edit: As François Brunault correctly points out, this computation is only feasible for instances in which the polynomials $F$ and common zeros $S$ can be represented in a computer. I am primarily interested in cases where the coefficients of the polynomials in $F$ are all rational and the entries of the members of $S$ are all algebraic. Presumably, techniques used in these cases can be extended to other computer-friendly cases.