Algorithms for quadratic forms over global function fields of odd characteristic

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.

Pythagoras Number Theory and the Scientific Beauty of the Theory Origin

In the long journey of the western aesthetics development, science and aesthetics especially the mathematics and the aesthetics relationship is more intimate. In the ancient Greece civilization, there is a very prominent characteristic, which is a kind of unparalleled thick interesting to question closely to the nature. As they don’t satisfied only stay in the imagination of the world picture view of the poetic in the mythology, and try to use the reason to guide the heaven sent human being’s imagination, they try eagerly to explain the nature order, evolution and movement. When the ancient Greece natural philosophers do the nature science research, they also do the aesthetic research. Pythagoras, as a mathematician and physicist and a astronomer, who with most of the disciples used the nature science view to see the aesthetics problems once at the beginning, which created the second to none of the method of quantitative research of the things.

HIGHER PRODUCT PYTHAGORAS NUMBERS OF SKEW FIELDS

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.