Pythagoras numbers of fields

A field of characteristic is said to have finite Pythagoras number if there exists an integer such that each nonzero sum of squares in can be written as a sum of squares, in which case the Pythagoras number of is defined to be the least such integer. As a consequence of Pfister's results on the level of fields, of a nonformally real field is always of the form or , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form , , and can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer there exists a formally real field with . As a refinement, we will show that if and are integers such that , then there exists a uniquely ordered field with and (resp. ), where (resp. ) denotes the supremum of the dimensions of anisotropic forms over which are torsion in the Witt ring of (resp. which are indefinite with respect to each ordering on ).

     

组合数集的毕达哥斯定理

本文得到两个组合数集的毕达哥斯定理的推广。(ⅰ )当n为奇数时∑[(n+ 3 ) / 2 ]t=0n +3-tt2 - ∑[(n+ 1) / 2 ]t=0n +1-tt2 2 +2 ∑[n/ 2 ]t=0n -tt · ∑[(n+ 4 ) / 2 ]t=0n +4-tt2 +4∑[(n+ 2 ) / 2 ]t=0n +2 -tt2= ∑[(n+ 1) /2 ]t=0n+ 1 -tt2 + ∑[(n+ 3) /2 ]t=0n+ 3-tt2 2 。(ⅱ )当n为偶数时∑[(n+ 4 ) / 2 ]t=0n+4-tt2 - ∑[n/ 2 ]t=0n-tt2 2 +2 ∑[(n+ 1) / 2 ]t=0n+1-tt · ∑[(n+ 3 ) / 2 ]t=0n+3-tt2 +4∑[(n+ 2 ) / 2 ]t=0n+2 -tt2= ∑[n/2 ]t=0n -tt2 + ∑[(n+ 4) /2 ]t=0n + 4 -tt2 2 。     

Looking for Pythagoras between the folds

The purpose of this short paper is to describe a new proof of the Pythagorean Theorem that involves paper folding.     

On the Pythagoras numbers of real analytic curves

We show that the Pythagoras number of a real analytic curve is the supremum of the Pythagoras numbers of its singularities, or that supremum plus 1. This includes cases when the Pythagoras number is infinite.      

Analytic surface germs with minimal Pythagoras number

We determine all complete intersection surface germs whose Pythagoras number is 2, and find that they are all embedded in ³ and have the property that every positive semidefinite analytic function germ is a sum of squares of analytic function germs. In addition, we discuss completely these properties for mixed surface germs in ³. Finally, we find in higher embedding dimension three different families with these same properties. Partially supported by DGICYT, BFM2002-04797 and HPRN-CT-2001-00271 Mathematical Subject Classification (2000): 11E25, 14P15.An erratum to this article can be found at     

Analysis of the two-fraction method for generating primitive Pythagoras triples

Finding methods for generating Pythagorean triples have been of great interest to Mathematicians since the Babylonians (from 1900 to 1600 BC). One of these methods is the less known two-fraction method which works for any two fractions whose product is 2. In this work, we analyse the method and show that it can be obtained from the fact that the excess of a right-angled triangle is even. Using this property, we develop several formulae for generating Primitive Pythagorean Triples.