By T. Husain
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12), we have Ap (ξ) = ξi1 ···ip ei1 ∧ · · · ∧ eip ∈ p V, and hence it is easy to show that |Ap (ξ)|v ≤ ςv,p! |ξ|v,⊗ , where the elementary inequality (a1 + · · · + an )2 ≤ n(a21 + · · · + a2n ) (ai ∈ R+ ) is used for the proof of the Archimedean case. In particular, if ξ ∈ Ap (ξ) = ξ. We can obtain the equality |ξ|v = cp |ξ|v,⊗ , where √ cp = Further, if η ∈ p! |v p V , then if v is Archimedean, if v is non-Archimedean. |ξ ⊗ η|v,⊗ = p+q p 1 2 |ξ|v |η|v if v is Archimedean. q! i1 <··· 59, it follows that values of the function ω become small if u is large. Hence Nochka weight is a gauge of a subgeneral position leaving a general position. Nochka’s original paper (see [299],[300],[301]) on the weights of Nochka was quite sketchy; a complete proof can be found in Chen’s thesis [56] (or see Fujimoto [107], Hu and Yang [176]). Here we omit the proof since it is very long. Let A = {a0 , a1 , . . , aq } (n ≤ u ≤ q) be in u-subgeneral position. Define the gauge Γ(A ) of A on a valuation v of κ by Γ(A ) = 1 ςv,(n+1)! Take a positive integer d. Let Jd be the permutation group on Z[1, d] and let ⊗d V be the d-fold tensor product of V . For each λ ∈ Jd , a linear isomorphism λ : ⊗d V −→ ⊗d V is uniquely defined by λ(ξ1 ⊗ · · · ⊗ ξd ) = ξλ−1 (1) ⊗ · · · ⊗ ξλ−1 (d) , ξj ∈ V (j = 1, . . , d). A vector ξ ∈ ⊗d V is said to be symmetric if λ(ξ) = ξ for all λ ∈ Jd . The set of all symmetric vectors in ⊗d V is a linear subspace of ⊗d V , denoted by d V , called the d-fold symmetric tensor product of V . Then dV dim n+d .