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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'' ∈ ''S''. * ''p'' is non-negative on ''S'' if ''p''(''x'') ≥ 0 for every ''x'' ∈ ''S''. * ''p'' is zero on ''S'' if ''p''(''x'') = 0 for every ''x'' ∈ ''S''. For certain sets ''S'', there exist algebraic descriptions of all polynomials that are positive (resp. non-negative, zero) on ''S''. Any such description is called a positivstellensatz (resp. nichtnegativstellensatz, nullstellensatz.) ==Examples== * Globally positive polynomials * * Every real polynomial in one variable is non-negative on ℝ if and only if it is a sum of two squares of real polynomials in one variable. * * The Motzkin polynomial ''X''4Y2 + ''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().〔T. S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224.〕 * * 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〔E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85-99.〕) * * Suppose that ''p'' ∈ ℝ() 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().〔B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.〕 * Polynomials positive on polytopes. * * For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If ''f,g1,...,gk'' have degree ≤ 1 and f(x) ≥ 0 for every x ∈ ℝ''n'' satisfying g1(x) ≥ 0,...,gk(x) ≥ 0, then there exist non-negative real numbers c0,c1,...,ck such that f=c0+c1g1+...+ckgk. * * Pólya's theorem:〔G. Pólya, Über positive Darstellung von Polynomen Vierteljschr, Naturforsch. Ges. Zürich 73 (1928) 141--145, in: R.P. Boas (Ed.), Collected Papers Vol. 2, MIT Press, Cambridge, MA, 1974, pp. 309--313〕 If ''p'' ∈ ℝ() is homogeneous and p is positive on the set , then there exists an integer ''m'' such that ''(x1+...+xn)m p'' has non-negative coefficients. * * Handelman's theorem:〔D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132 (1988), no. 1, 35--62.〕 If ''K'' is a compact polytope in Euclidean ''d''-space, defined by linear inequalities gi ≥ 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,〔K. Schmüdgen, The ''K''-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203–206.〕〔T. Wörmann Strikt Positive Polynome in der Semialgebraischen Geometrie, Univ. Dortmund 1998.〕 Putinar's positivstellensatz〔M. Putinar, Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (1993), no. 3, 969–984.〕〔T. Jacobi, A representation theorem for certain partially ordered commutative rings. Math. Z. 237 (2001), no. 2, 259–273.〕 and Vasilescu's positivstellensatz.〔Vasilescu, F.-H. Spectral measures and moment problems. Spectral analysis and its applications, 173--215, Theta Ser. Adv. Math., 2, Theta, Bucharest, 2003. See Theorem 1.3.1.〕 The point here is that no denominators are needed. * * For nice compact semialgebraic sets of low dimension there exists a nichtnegativstellensatz without denominators.〔C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039–1069.〕〔C. Scheiderer, Sums of squares on real algebraic curves. Math. Z. 245 (2003), no. 4, 725–760.〕〔C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta Math. 119 (2006), no. 4, 395–410.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Positive polynomial」の詳細全文を読む スポンサード リンク
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