Examples of the Stufe and pythagoras number of fields using the Hasse-Minkowski theorem

Squares ◽  
1993 ◽  
pp. 259-271
Keyword(s):  
Minkowski Theorem ◽  
Pythagoras Number

scholarly journals HIGHER PRODUCT PYTHAGORAS NUMBERS OF SKEW FIELDS

2010 ◽  
Vol 03 (01) ◽  
pp. 193-207
Author(s):  
Dejan Velušček
Keyword(s):  
Laurent Series ◽  
Skew Field ◽  
Skew Fields ◽  
Pythagoras Number

We introduce the n–th product Pythagoras number p n(D), the skew field analogue of the n–th Pythagoras number of a field. For a valued skew field (D, v) where v has the property of preserving sums of permuted products of n–th powers when passing to the residue skew field k v and where Newton's lemma holds for polynomials of the form Xn - a, a ∈ 1 + I v , p n(D) is bounded above by either p n( k v ) or p n( k v ) + 1. Spherical completeness of a valued skew field (D, v) implies that the Newton's lemma holds for Xn - a, a ∈ 1 + I v but the lemma does not hold for arbitrary polynomials. Using the above results we deduce that p n (D((G))) = p n(D) for skew fields of generalized Laurent series.

Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional

2014 ◽  
Vol 66 (4) ◽  
pp. 783-825 ◽  
Author(s):  
Ivan Izmestiev
Keyword(s):  
Existence And Uniqueness ◽  
Convex Polyhedra ◽  
Future Research ◽  
Second Derivative ◽  
De Sitter ◽  
Spherical Space ◽  
Infinitesimal Rigidity ◽  
Minkowski Theorem ◽  
Perfect Duality ◽  
Derivatives Of

Abstract. The paper presents a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert–Einstein functional on the space of “warped polyhedra” with a fixed metric on the boundary.The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.In the spherical space and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert–Einstein functional and the volume, as well as between both kinds of rigidity.We review some of the related work and discuss directions for future research.

A Nonsymmetric Hasse-Minkowski Theorem

1977 ◽  
Vol 99 (4) ◽  
pp. 755 ◽  
Author(s):  
William C. Waterhouse
Keyword(s):  
Minkowski Theorem

scholarly journals Algorithms for quadratic forms over global function fields of odd characteristic

2021 ◽  
Vol 55 (3) ◽  
pp. 68-72
Author(s):  
Mawunyo Kofi Darkey-Mensah
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Function Fields ◽  
Number Fields ◽  
Global Function ◽  
Global Function Fields ◽  
Anisotropic Part ◽  
Pythagoras Number ◽  
Present Algorithm

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.

scholarly journals Estimates of the Pythagoras number of Rm[x1,…,xn] through lattice points and polytopes

2008 ◽  
Vol 308 (23) ◽  
pp. 5771-5781 ◽  
Author(s):  
David B. Leep ◽  
Colin L. Starr
Keyword(s):  
Lattice Points ◽  
Pythagoras Number
Sign in / Sign up

Export Citation Format

Share Document