On Witt rings of function fields of real analytic surfaces and curves.

Let y be a paracompact connected real analytie manifoid of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold ~c of the sanie dimension. Moreover by a prime divisor of y we shell mean the irreducible germ alang y of a cadimension ono subvariety of vc wbich le an invariant of tite cemplex conjugation. This notion la independent of the chaice of the complexification ~ In tbe one-dimensional case prime divisora are just pointa, in tbe twa-dimensional- analytic curves or elliptic points (intersectione of twa conjugatedcomplex analytic curves). Every such divisor induces a discreto valuatien on the field M of meramorphic functions an V-the arder of the zero or minus the erder of the polo of the function. Therefore it induces tire so called residue homomorphisms (first and seeand) ef tire Witt greup of tire fleid M to the Witt graup of tire residuo fleld- tire funetian fleld of tire divisor. Tire main goal of tbk paper la te shaw that tbe intersection of kernels of alí second residuo homomarpirisme ansociated to prime divisors is isamorphic te tire Witt group of tire Rioniamijan bundies an 1’. As en example of an applicatian of tiria result wo previde tho new proof of the Artun-Lang praperty for ano and two dimensional real analytic manifolds (batir cempact and noncompact), which is neitber based on the deseriptien of all posible ordorings of tire fiold of meromerpitie fimetione nor en tire cexnpactificatian of tite variety