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Characterization of Classes of Polynomial Functions


In this paper, some classes of local polynomial functions on abelian groups are characterized by the properties of their variety. For this characterization, we introduce a numerical quantity depending on the variety of the local polynomial only. Moreover, we show that the known characterization of polynomials among generalized polynomials can be simplified: a generalized polynomial is a polynomial if and only if its variety contains finitely many linearly independent additive functions.

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  1. Djokovič, D.Ž.: A representation theorem for (X 1 − 1)(X 2 − 1) . . . (X n − 1) and its applications. Ann. Pol. Math. 22, 189–198 (1969/1970)

  2. Fréchet M.: Une definition fonctionelle des polynomes. Nouv. Ann. 9, 145– 162 (1909)

    Google Scholar 

  3. Laczkovich M.: Polynomial mappings on abelian groups. Aequ. Math. 68(3), 177–199 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Laczkovich, M.: Local spectral synthesis on abelian groups. Acta Math. Hung. 143(2), 313–329 (2014)

  5. Mazur S., Orlicz W.: Grundlegende Eigenschaften der Polynomischen Operationen I. Stud. Math. 5, 50–68 (1934)

    MATH  Google Scholar 

  6. Prager W., Schwaiger J.: Generalized polynomials in one and in several variables. Math. Pannon. 20(2), 189–208 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Reich, L., Schwaiger, J.: On polynomials in additive and multiplicative functions. In: Functional equations: history, applications and theory, Math. Appl., pp. 127–160. Reidel, Dordrecht (1984)

  8. Schwaiger J., Prager W.: Polynomials in additive functions and generalized polynomials. Demonstr. Math. 41(3), 589–613 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Székelyhidi, L.: On Fréchet’s functional equation. Monatsh. für Math. to appear (2014). doi:10.1007/500605-013-0-590-2

  10. Székelyhidi, L.: Convolution type functional equations on topological abelian groups. World Scientific Publishing Co. Inc., Teaneck, NJ (1991)

  11. Székelyhidi L.: Polynomial functions and spectral synthesis. Aequ. Math. 70(1–2), 122–130 (2005)

    Article  MATH  Google Scholar 

  12. Székelyhidi L.: Noetherian rings of polynomial functions on Abelian groups. Aequationes Math 84(1–2), 41–50 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lijn G.: La définition fonctionnelle des polynômes dans les groupes abéliens. Fund. Math 33, 42–50 (1939)

    MathSciNet  Google Scholar 

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Correspondence to J. M. Almira.

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The research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. NK-81402.

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Almira, J.M., Székelyhidi, L. Characterization of Classes of Polynomial Functions. Mediterr. J. Math. 13, 301–307 (2016).

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