Homogeneous tri-additive forms and derivations |
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Authors: | Bruce Ebanks CT Ng |
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Institution: | a Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, MS 39762, United States b Department of Pure Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 |
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Abstract: | Gleason A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ2Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ2S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V3 into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ3T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation |
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Keywords: | 11E76 15A21 39B52 39B72 |
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