Titlebook: A Course in Real Algebraic Geometry; Positivity and Sums Claus Scheiderer Textbook 2024 The Editor(s) (if applicable) and The Author(s), u

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The Role of the European Union,s are given to polynomials that are strictly positive on some domain, such as the positivstellensätze of Schmüdgen and Putinar. An optional alternative approach is offered, which uses pure states for convex cones and leads to the Archimedean local-global principle.

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https://doi.org/10.1007/978-1-349-20902-6ariety . has minimal degree if, and only if, every non-negative quadratic form on . is a sum of squares of linear forms. Quantitative refinements are given as well. These results encompass several major classical theorems, among them the Hilbert 1888 theorems that were discussed in Chapter 2.

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Positive Polynomials with Zeros,local-global principle, which is given an independent second proof. Combining this result with an analysis of (weighted) sums of squares in local rings, a series of existence and non-existence results is obtained for sums of squares representations of non-negative polynomials.

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Sums of Squares on Projective Varieties,ariety . has minimal degree if, and only if, every non-negative quadratic form on . is a sum of squares of linear forms. Quantitative refinements are given as well. These results encompass several major classical theorems, among them the Hilbert 1888 theorems that were discussed in Chapter 2.

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