/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.

https://hal.archives-ouvertes.fr/hal-00609687/document

Real Algebraic Geometry

This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4 For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras

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The Book of Involutions

We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

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Sums of Squares, Moment Matrices and Optimization Over Polynomials

Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dimV ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if V is a smooth surface with only real divisors at infinity. The “compact” case is harder. We completely settle the case of smooth curves of genus ≤ 1: If such a curve has a complex point at infinity, then every psd function is sos, provided the field R is archimedean. If R is not archimedean, there are counter-examples of genus 1.

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Sums of squares of regular functions on real algebraic varieties

This paper gives a survey, with detailed references to the literature, on recent developments in real algebra and geometry concerning the polarity between positivity and sums of squares. After a review of foundational material, the topics covered are Positiv- and Nichtnegativstellensatze, local rings, Pythagoras numbers, and applications to moment problems.

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Positivity and sums of squares: A guide to recent results

Given an affine algebraic variety V over ℝ with real points V(ℝ) compact and a non-negative polynomial function f∈ℝ[V] with finitely many real zeros, we establish a local-global criterion for f to be a sum of squares in ℝ[V]. We then specialize to the case where V is a curve. The notion of virtual compactness is introduced, and it is shown that in the local-global principle, compactness of V(ℝ) can be relaxed to virtual compactness. The irreducible curves on which every non-negative polynomial is a sum of squares are classified. All results are extended to the more general framework of preorders. Moreover, applications to the K-moment problem from analysis are given. In particular, Schmudgen’s solution of the K-moment problem for compact K is extended, for dim (K)=1, to the case when K is virtually compact.

Sums of squares on real algebraic curves

Example 4: The assertion for n ≤ 2 is true. Also, for odd n ≥ 3, the assertion is true, since in this case the curve C has exactly one point at infinity, which is real. If n ≥ 4 is even (and C is real), there are two points at infinity which are both real. Therefore, the moment problem for K is not finitely solvable if K is unbounded on two half-branches of C(R) at infinity which represent different points at infinity. Otherwise, the moment problem for K is finitely solvable. More concretely, this means the following for n ≥ 4. Let Q1, Q2, Q3, Q4 be the four quadrants of the real plane (numbered counter-clockwise in the usual way). If n ≡ 0 (mod 4), the moment problem for K is finitely solvable iff at least one of K ∩ (Q1 ∪Q2), K ∩ (Q3 ∪Q4) is bounded. If n ≡ 2 (mod 4), the moment problem for K is finitely solvable iff at least one of K ∩ (Q1 ∪Q3), K ∩ (Q2 ∪Q4) is bounded.

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The moment problem for non-compact semialgebraic sets

/pdf/einfuhrung-in-die-reelle-algebra-1lq11e4dan.pdf

Claus Scheiderer

Papers

Sums of squares of regular functions on real algebraic varieties

Positivity and sums of squares: A guide to recent results

Sums of squares on real algebraic curves

The moment problem for non-compact semialgebraic sets

Einführung in die reelle Algebra