Real algebraic geometry

Positive polynomial

In mathematics, a positive polynomial on a particular set is a polynomial whose values are positive on that set. Let p be a polynomial in n variables with real coefficients and let S be a subset of the n-dimensional Euclidean space ℝn. We say that: * p is positive on S if p(x) > 0 for every x in S. * p is non-negative on S if p(x) ≥ 0 for every x in S. * p is zero on S if p(x) = 0 for every x in S. For certain sets S, there exist algebraic descriptions of all polynomials that are positive, non-negative, or zero on S. Such a description is a positivstellensatz, nichtnegativstellensatz, or nullstellensatz. This article will focus on the former two descriptions. For the latter, see Hilbert's Nullstellensatz for the most known nullstellensatz. (Wikipedia).

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Related pages

Polytope | Subset | Hilbert's seventeenth problem | Hilbert's Nullstellensatz | Polynomial | If and only if | Mathematics | Set (mathematics) | Coefficient | Integer | Real number | Trigonometric polynomial | Euclidean space | Degree of a polynomial | Semialgebraic set | Matrix polynomial | Homogeneous polynomial | Polynomial SOS