A classical problem in the theory of quadratic forms is to decide whether two given integral quadratic forms are equivalent. That is, whether their symmetric integral matrices A and B satisfy A=X’BX for some unimodular integral matrix X. For definite forms one can construct a simple decision procedure. Somewhat surprisingly, no such procedure was known for indefinite forms until the work of C. L. Siegel in the early 1970s. In the late 1990s, D. W. Masser made the following conjecture related to this problem: Let n be at least 3, and suppose A, B are equivalent. Then there exists a unimodular integral matrix X such that A=X’BX and ||X||< C(||A||+||B||)^k, where the constants C, k depend only on the dimension n. In this talk we shall discuss our recent resolution of this conjecture based on a joint work with Professor Gregory A. Margulis.

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