In algebraic geometry, the Nullstellensatz gives an algebraic characterization of when a multivariate polynomial vanishes on a set of points defined by a system of (complex) polynomial equations.

I’ve just come across an analogue of this over the reals. In this case, it’s a purely algebraic characterization of when a multivariate polynomial is guaranteed to be positive over a set of points defined by a system of polynomial inequalities. In analogy with the complex case, the result is known as the Positivstellensatz.