It’s usually easier to show that a problem has a solution than to show that it does not have a solution.
Analogy with prime numbers
Showing that a number is prime amounts to saying that the problem of finding nontrivial factors has no solution. How could you convince a skeptic that a large number N is prime? If N is small enough that it is possible to divide N by every integer less than √N then that would do it, but that’s not practical if, say, N has 100 digits. Nevertheless there is a way.
Primality certificates are a way to prove that a number is prime, just as demonstrating a factor is a way to prove that a number is not prime. A primality certificate is a set of calculations a skeptic can run that will prove that the number in question is prime. Producing such a certificate takes a lot of computation, but verifying the certificate does not.
Analogy with convex optimization
Claiming that you’ve found the maximum value of some function amounts to saying a certain problem has no solution, namely the problem of finding where the function takes on a larger value than the one you claim is optimal. How can you convince a skeptic that you’ve really found the maximum?
You could say “I’ve tried really hard. Here’s a list of things I’ve tried, and see, none of them give a larger value.” Sometimes that’s all you can do. But for convex optimization problems, you can produce a certificate that a skeptic could evaluate that proves that your solution is optimal. More on that here.
Systems of polynomial equations
Now suppose you have a system of polynomial equations. If you’re able to find a solution, then this solution is its own certificate. Want to prove that supposed solution actually is a solution? Plug it in and see.
But how do you prove that you haven’t overlooked any possibilities and there simply isn’t a solution to be found? Once again there is a kind of certificate.
Suppose you have a system of m polynomial equations in n complex variables.
The system of equations has a solution if and only if the algebraic variety generated by the f‘s is non-empty.
If we can find a polynomial combination of the f‘s that equals 1, i.e. we can find g1 through gm such that
then the variety generated by the f‘s is empty. Such a polynomial combination would be a certificate that the system of equations has no common solution, a certificate of infeasibility. There is an algorithm for finding this polynomial combination, the extended Buchberger algorithm, an analog of the extended Euclidean algorithm.
There is a corresponding theorem for polynomials of a real variable, the so-called positivstellensatz, but it is a little more complicated. The name “complex numbers” is unfortunate because passing to “complex” numbers often simplifies problems.