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Characterizing ring derivations of all orders via functional equations: results and open problems

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Abstract

We provide a unifying framework for the treatment of equations of the form

$$\sum_{k=1}^n x^{p_k} f_k (x^{q_k}) = 0$$

for additive maps f k and integers p k , q k (1 ≤  k ≤  n). We show how to solve many equations of this type, and we present some open problems. In general our unknown functions map an integral domain of characteristic zero into itself. When negative exponents appear, we restrict our attention to fields of characteristic zero. All of the results could be formulated for integral domains or fields of sufficiently large characteristic as well.

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References

  1. Aczél J., Daróczy Z.: On Measures of Information and Their Characterizations. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Ebanks B.: On the equation F(X) + M(X)G(X −1) = 0 on K n. Linear Algebra Appl. 125, 1–17 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ebanks B., Ng C.T.: Homogeneous tri-additive forms and derivations. Linear Algebra Appl. 435, 2731–2755 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ebanks B., Sahoo P., Sander W.: Characterizations of Information Measures. World Scientific, Singapore (1998)

    Book  Google Scholar 

  5. Eisenbud D.: Commutative Algebra: with a View Toward Algebraic Geometry. Springer, New York (1999)

    Google Scholar 

  6. Gleason A.M.: The definition of a quadratic form. Am. Math. Monthly 73, 1049–1066 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  7. Halter-Koch F.: Characterization of field homomorphisms and derivations by functional equations. Aequat. Math. 59, 298–305 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Halter-Koch F., Reich L.: Charakterisierung von Derivationen höherer Ordnung mittels Funktionalgleichungen. Österreich. Akad. Wiss. Math. Natur. Kl. Sitzungsber. II 207, 123–131 (1998)

    MATH  MathSciNet  Google Scholar 

  9. Hartshorne R.: Algebraic Geometry. Springer, New York (1997)

    Google Scholar 

  10. Kannappan Pl., Kurepa S.: Some relations between additive functions-I. Aequat. Math. 4, 163–175 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. In:Cauchy’s Equation and Jensen’s Inequality. Birkhäuser, Basel (2009)

  12. Kurepa S.: The Cauchy functional equation and scalar product in vector spaces. Glasnik Mat.-Fiz. Astronom. Ser. II Drutvo Mat. Fiz. Hrvatske 19, 23–36 (1964)

    MATH  MathSciNet  Google Scholar 

  13. Kurepa S.: Remarks on the Cauchy functional equation. Publ. Inst. Math. (Beograd) (N.S.) 5(19), 85–88 (1965)

    MathSciNet  Google Scholar 

  14. Maksa Gy.: The general solution of a functional equation related to the mixed theory of information. Aequat. Math. 22, 90–96 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ng C.T.: On a generalized fundamental equation of information. Can. J. Math. 35, 862–872 (1983)

    Article  MATH  Google Scholar 

  16. Ng C.T.: On the equation F(x) + M(x)G(1/x) = 0 and homogeneous bi-additive forms. Linear Algebra Appl. 93, 255–279 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nishiyama A., Horinouchi S.: On a system of functional equations. Aequat. Math. 1, 1–5 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  18. Reich L.: Derivationen zweiter Ordnung als Lösungen von Funktionalgleichungen. Grazer Math. Ber. 337, 45–65 (1998)

    MATH  Google Scholar 

  19. Unger J., Reich L.: Derivationen höherer Ordnung als Lösungen von Funktionalgleichungen. Grazer Math. Ber. 336, 1–83 (1998)

    MATH  MathSciNet  Google Scholar 

  20. Zariski, O., Samuel, P.: Commutative Algebra, vol. 1. Springer, New York (1986)

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  1. Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, MS, 39762, USA

    Bruce Ebanks

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  1. Bruce Ebanks
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Correspondence to Bruce Ebanks.

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Ebanks, B. Characterizing ring derivations of all orders via functional equations: results and open problems. Aequat. Math. 89, 685–718 (2015). https://doi.org/10.1007/s00010-014-0256-8

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  • DOI: https://doi.org/10.1007/s00010-014-0256-8

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