A New Local-Global Principle for Quadratic Functional Fields

A new local global principle for quadratic functional fields has been reported. The problem of the existence of nontrivial units is difficult; it is deeply related to torsion in the Jacobian varieties of curves and to continuous fractions in function fields. The proof of this theorem 1 uses properties of the Jacobian variety of the curve and it localizations. Various equations and theorems have also been provided in support of the problem. An algorithmic solution of the problem of the existence of nontrivial units in the ring for polynomials with deg was obtained in Theorem 1 by reducing the problem under consideration to the problem of torsion in elliptic curves. Using the methods developed, one can give a complete answer to the question of for what polynomials the ring has nontrivial units. In the proof of Theorem 3, the calculation of the determinants, the factorization of polynomials, and the proof of the irreducibility over of the polynomials were performed by using the Maple computer algebra system.