On the Embedding of an Affine Space into a Projective Space |
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Authors: | Hiroaki Taniguchi |
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Affiliation: | (1) 72-3-202 Kounami, Wakayama City, 649-6315, Japan |
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Abstract: | Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x1,x2,...,xn) = [(1,x1 ,x2 ,...,xn)B], where the right-hand side is the equivalence class of (1,x1 ,x2,...,xn)B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dimkK, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim kK=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd. |
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Keywords: | embedding affine space |
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