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Functional Equation Related to Quadratic and Norm Forms


Let T(a 1, a 2,..., a n ) be a norm form of some finite proper extension of ℚ in a certain fixed integral basis, or even some integer form satisfying certain conditions. We are interested in two problems. One is finding all functions f : ℤ → ℂ satisfying the integer functional equation f(T(a 1, a 2,..., a n )) = T(f(a 1), f(a 2),..., f(a n )). Another closely related problem is finding all functions f : ℤ → ℂ such that T(f(a 1, f(a 2,..., f(a n )) depends only on the value of T(a 1, a 2,..., a n ).The second question was studied before for special quadratic forms. We extend these investigations to other types of quadratic forms and, thus, partially solve the second problem for them. The solution of the first problem for one cubic field is also presented. Finally, we give the corresponding conjecture for the first problem and, additionally, several remarks concerning the choice of norm forms.

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Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 153–172, April–June, 2005.

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Alkauskas, G. Functional Equation Related to Quadratic and Norm Forms. Lith Math J 45, 123–141 (2005).

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  • norm form
  • square additive functions
  • integer functional equation