Abstract
Let k be a field of characteristic not two. It is proved that if a quadratic k[X1,...,Xn]-space is a hyperbolic space under an extension to the field ofrational functions k(X1,...,Xn, then the initial space is also hyperbolic. Earlier this result was obtained by Ojanguren under the additional assumption that k is an infinite perfect field (of characteristic not two).
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References
M. Ojanguren, “Quadratic forms over regular rings,” J. Indian Math. Soc.,44, 109–116 (1979).
T.-Y. Lam, Serre's Conjecture, Lecture Notes in Mathematics635, Springer, New York (1978).
V. I. Kopeiko and A. A. Suslin, “Quadratic modules over polynomial rings,” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklov. (LOMI),86, 114–124 (1979).
T.-Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin, Reading, Massachusetts (1973).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 124–125, 1991.
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Kopeiko, V.I. On a theorem of Ojanguren. J Math Sci 63, 683 (1993). https://doi.org/10.1007/BF01097982
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DOI: https://doi.org/10.1007/BF01097982