A short proof of alienation of additivity from quadraticity

Year, pages: 2019, 57 - 62

Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional equation \[f(x+y) + g(x+y) + g(x-y) = f(x) + f(y) + 2g(x) + 2g(y)\] forces the unknown functions $f$ and $g$ to be additive and quadratic, respectively, modulo a constant. \par Motivated by the observation that the equation \[f(x+y) + f(x²) = f(x) + f(y) + f(x)²\] implies both the additivity and multiplicativity of $f$, we deal also with the alienation phenomenon of equations in a single and several variables.

Ger, R. 2019. A short proof of alienation of additivity from quadraticity. In Tatra Mountains Mathematical Publications, vol. 74, no.2, pp. 57-62. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2019-0019

Ger, R. (2019). A short proof of alienation of additivity from quadraticity. Tatra Mountains Mathematical Publications, 74(2), 57-62. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2019-0019

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava

Licensed under the Creative Commons Attribution-NC-ND4.0 International Public License.