In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively non-negative) on that set. Precisely, Let p be a polynomial in n variables with real coefficients and let S be a subset of the n-dimensional Euclidean space. We say that:

Positivstellensatz (and nichtnegativstellensatz)

For certain sets S, there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on S. Such a description is a positivstellensatz (resp. nichtnegativstellensatz). The importance of Positivstellensatz theorems in computation arises from its ability to transform problems of polynomial optimization into semidefinite programming problems, which can be efficiently solved using convex optimization techniques.

Examples of positivstellensatz (and nichtnegativstellensatz)

Generalizations of positivstellensatz

Positivstellensatz also exist for signomials, trigonometric polynomials, polynomial matrices, polynomials in free variables, quantum polynomials, and definable functions on o-minimal structures.