scholarly journals Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

2012 ◽  
Vol 62 (6) ◽  
pp. 2131-2143 ◽  
Author(s):  
Yong HU
Keyword(s):  
Quadratic Forms ◽  
Two Dimensional ◽  
Global Principle

scholarly journals Applied Optimization and Approximation

2014 ◽  
Vol 3 (4) ◽  
pp. 37-48
Author(s):  
Octav Olteanu
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Linear Function ◽  
Polynomial Approximation ◽  
Moment Problem ◽  
Quadratic Forms ◽  
Two Dimensional ◽  
Several Variables ◽  
Applied Optimization ◽  
The Moment

The present work deals with optimization in kinematics, generalizing previous results of the author. A second theme is maximizing the constrained gain linear function and minimizing the constrained cost function. Elementary notions of optimal control are considered as well. Finally, polynomial approximation results on unbounded subsets in several variables are applied to the moment problem. The existence of the solution of a two dimensional moment problem is characterized in terms of quadratic forms.

A local-global principle for quadratic forms

1975 ◽  
Vol 142 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Alexander Prestel
Keyword(s):  
Quadratic Forms ◽  
Global Principle

scholarly journals Moment Problems on Bounded and Unbounded Domains

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Octav Olteanu
Keyword(s):  
Quadratic Forms ◽  
Necessary Conditions ◽  
Unbounded Domains ◽  
Continuous Functions ◽  
Moment Problems ◽  
Two Dimensional ◽  
Space Of Continuous Functions ◽  
One Dimensional ◽  
Sufficient And Necessary Conditions ◽  
The One

Using approximation results, we characterize the existence of the solution for a two-dimensional moment problem in the first quadrant, in terms of quadratic forms, similar to the one-dimensional case. For the bounded domain case, one considers a space of complex analytic functions in a disk and a space of continuous functions on a compact interval. The latter result seems to give sufficient (and necessary) conditions for the existence of a multiplicative solution.

scholarly journals Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

2009 ◽  
Vol 145 (2) ◽  
pp. 309-363 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Fei Xu
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Integral Quadratic Form ◽  
Linear Algebraic Groups ◽  
Global Principle

AbstractAn integer may be represented by a quadratic form over each ring ofp-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

scholarly journals A local global principle for quadratic forms over polynomial rings

1982 ◽  
Vol 74 (1) ◽  
pp. 264-269 ◽  
Author(s):  
Raman Parimala ◽  
R Sridharan
Keyword(s):  
Quadratic Forms ◽  
Polynomial Rings ◽  
Global Principle

scholarly journals Spinor groups with good reduction

2019 ◽  
Vol 155 (3) ◽  
pp. 484-527 ◽  
Author(s):  
Vladimir I. Chernousov ◽  
Andrei S. Rapinchuk ◽  
Igor A. Rapinchuk
Keyword(s):  
Quadratic Forms ◽  
Galois Cohomology ◽  
Unitary Groups ◽  
Global Field ◽  
Two Dimensional ◽  
Good Reduction ◽  
Isomorphism Classes ◽  
Unramified Cohomology ◽  
Finiteness Properties ◽  
Local Map

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$. We show that for a given $n\geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\mathsf{G}_{2}$.

scholarly journals Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Diego Izquierdo ◽  
Giancarlo Lucchini Arteche
Keyword(s):  
Laurent Series ◽  
Homogeneous Spaces ◽  
Function Fields ◽  
Independent Interest ◽  
Complex Numbers ◽  
Brauer Group ◽  
Two Dimensional ◽  
Global Fields ◽  
Global Principles ◽  
Global Principle

Abstract In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field ℂ ⁢ ( ( x , y ) ) {\mathbb{C}((x,y))} of Laurent series in two variables over the complex numbers and over function fields of curves over ℂ ⁢ ( ( t ) ) {\mathbb{C}((t))} . We give examples that prove that the Brauer–Manin obstruction with respect to the whole Brauer group is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.

scholarly journals Simultaneous integer values of pairs of quadratic forms

2017 ◽  
Vol 2017 (727) ◽  
Author(s):  
D. R. Heath-Brown ◽  
L. B. Pierce
Keyword(s):  
Quadratic Forms ◽  
Circle Method ◽  
Two Dimensional ◽  
Local Conditions ◽  
Dimensional Version ◽  
Almost All

AbstractWe prove that a pair of integral quadratic forms in five or more variables will simultaneously represent “almost all” pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.

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