Definite Quadratic Forms over F_q[x]

Le R be a principalidec domain withquotie tfie F.An R-lattice is afre R-module offinite rank spanninganinne productspace ove .The classififi problem asks for areg- ee e se ofcrite to degwhe two give R-lattice are isomee that is, whethe is an inne ductpreg-gege carryingone lattice ontothe othe In this pap R isthe polynomial ring Fq [x],whe q is afinite fiet of oddorde q.For q [x]-lattice as for Z-lattice the theti splits into"deg and"indeg- cased and this pape seg the classificationproble inthe de case The claswx;3T""k of definite quadraticforms over the rational integers is a notorious" intractable problem. An exceptionis the binarycasr Gaus ss wed that every definite binary form over equivalent to a unique "reduced" form that can be found algorithmically; and two binary forms are equivalent if and only if they have the se3 reduced form. But forforms 3, while there are a number of reductiontheories developed by MinkowsW andothers nonehas proved entirelysely3"kFFk3 . For example, a given form may be equivalent to more than one reduced form; and determining whether two given reducedforms are equivalent may be computationally daunting. We refer the reader to Nipp [11] for a concrete expos3w;; ofthes matters and to Conway--Sloane [2], Chapter 15, for a broadsoa ey of theclas""F3Tw[" problem over Z