Systems of Diagonal Equations Over p-Adic Fields |
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Authors: | Knapp Michael P. |
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Affiliation: | Department of Mathematics, University of Michigan 525 East University Avenue, Ann Arbor, MI 48109-1109, USA |
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Abstract: | Let K be a p-adic field, and consider the system F = (F1,...,FR)of diagonal equations (1) with coefficients in K. It is an interesting problem in numbertheory to determine when such a system possesses a nontrivialK-rational solution. In particular, we define *(k, R, K) tobe the smallest natural number such that any system of R equationsof degree k in N variables with coefficients in K has a nontrivialK-rational solution provided only that N*(k, R, K). For example,when k = 1, ordinary linear algebra tells us that *(1, R, K)= R + 1 for any field K. We also define *(k, R) to be the smallestinteger N such that *(k, R, Qp) N for all primes p. |
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