Abstract. We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre's theorem of the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature and stems from numerical methods using semidefinite programming to simulate or control discrete-time behaviour of systems.
A recently found local-global principle for quadratic forms over function fields of curves over a complete discretely valued field is applied to the study of quadratic forms, sums of squares, and related field invariants.
We show that the transcendence degree of a real function field over an arbitrary real base field is a strict lower bound for its Pythagoras number and a weak lower bound for all its higher Pythagoras numbers.
Abstract. Given a geometrically unirational variety over an infinite base field, we show that every finite separable extension of the base field that splits the variety is the residue field of a closed point. As an application, we obtain a characterization of function fields of smooth conics in which every sum of squares is a sum of two squares.
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