Positivstellensatz

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 ℝ^''n''. 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.

Examples of positivstellensatz (and nichtnegativstellensatz)

* Globally positive polynomials and sum of squares decomposition.
** Every real polynomial in one variable and with even degree is non-negative on ℝ if and only if it is a sum of two squares of real ''polynomials'' in one variable. This equivalence does not generalize for polynomial with more than one variable: for instance, the Motzkin polynomial ''X''⁴''Y''² + ''X''²''Y''⁴ − 3''X''²''Y''² + 1 is non-negative on ℝ² but is not a sum of squares of elements from ℝ 'X'', ''Y''
** A real polynomial in ''n'' variables is non-negative on ℝ^''n'' if and only if it is a sum of squares of real ''rational'' functions in ''n'' variables (see Hilbert's seventeenth problem and Artin's solution).
** Suppose that ''p'' ∈ ℝ 'X''₁, ..., ''X''_''n''is homogeneous of even degree. If it is positive on ℝ^''n'' \ , then there exists an integer ''m'' such that (''X''₁² + ... + ''X''_''n''²)^''m'' ''p'' is a sum of squares of elements from ℝ 'X''₁, ..., ''X''_''n''
* Polynomials positive on polytopes.
** For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If ''f'', ''g''₁, ..., ''g''_''k'' have degree ≤ 1 and ''f''(''x'') ≥ 0 for every ''x'' ∈ ℝ^''n'' satisfying ''g''₁(''x'') ≥ 0, ..., ''g''_''k''(''x'') ≥ 0, then there exist non-negative real numbers ''c''₀, ''c''₁, ..., ''c''_''k'' such that ''f'' = ''c''₀ + ''c''₁''g''₁ + ... + ''c''_''k''''g''_''k''.
** Pólya's theorem: If ''p'' ∈ ℝ 'X''₁, ..., ''X''_''n''is homogeneous and ''p'' is positive on the set , then there exists an integer ''m'' such that (''x''₁ + ... + ''x''_''n'')^''m'' ''p'' has non-negative coefficients.
** Handelman's theorem: If ''K'' is a compact polytope in Euclidean ''d''-space, defined by linear inequalities ''g''_''i'' ≥ 0, and if ''f'' is a polynomial in ''d'' variables that is positive on ''K'', then ''f'' can be expressed as a linear combination with non-negative coefficients of products of members of .
* Polynomials positive on semialgebraic sets.
** The most general result is Stengle's Positivstellensatz.
** For compact semialgebraic sets we have Schmüdgen's positivstellensatz, Putinar's positivstellensatz and Vasilescu's positivstellensatz. The point here is that no denominators are needed.
** For nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators.

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.

References

* Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise. ''Real Algebraic Geometry''. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) esults in Mathematics and Related Areas (3) 36. Springer-Verlag, Berlin, 1998. x+430 pp. .
* Marshall, Murray. "Positive polynomials and sums of squares". ''Mathematical Surveys and Monographs'', 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. , .

Notes

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See also

* Polynomial SOS
*Hilbert's seventeenth problem

Real algebraic geometry