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Definite Quadratic Forms over Fq[x] Larry J. Gerstein

Summary: Definite Quadratic Forms over Fq[x]
Larry J. Gerstein
Department of Mathematics
University of California
Santa Barbara, CA 93106
E-mail: gerstein@math.ucsb.edu
Version: September 30, 2002
ABSTRACT. Let R be a principal ideal domain with quotient field F. An
R-lattice is a free R-module of finite rank spanning an inner product space over
F. The classification problem asks for a reasonably effective set of criteria
to determine when two given R-lattices are isometric; that is, when there is an
inner-product preserving isomorphism carrying one lattice onto the other. In this
paper R is the polynomial ring Fq[x], where Fq is a finite field of odd order q. For
Fq[x]-lattices as for Z-lattices the theory splits into "definite" and "indefinite"
cases, and this paper settles the classification problem in the definite case.
The classification of definite quadratic forms over the rational integers
is a notoriously intractable problem. An exception is the binary case:
Gauss showed that every definite binary form over Z is equivalent to a
unique "reduced" form that can be found algorithmically; and two binary
forms are equivalent if and only if they have the same reduced form. But


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics