On additive solutions of a linear equation

Varga, Adrienn

doi:10.1007/s10474-009-9148-0

Cited by 8 publications

(7 citation statements)

References 3 publications

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“…Then we have an eliminating technique due to A. Varga [8] to solve equation (A 1 ), i.e. we can decide algorithmically the problem of the existence of non-trivial solutions.…”

Section: The Case Of Algebraic Parametersmentioning

confidence: 99%

Nontrivial solutions of linear functional equations: methods and examples

Varga¹,

Vincze²

2015

OpMath

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Abstract. For a wide class of linear functional equations the solutions are generalized polynomials. The existence of non-trivial monomial terms of the solution strongly depends on the algebraic properties of some related families of parameters. As a continuation of the previous work [A. Varga, Cs. Vincze, G. Kiss, Algebraic methods for the solution of linear functional equations, Acta Math. Hungar.] we are going to present constructive algebraic methods of the solution in some special cases. Explicit examples will be also given.

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“…Then we have an eliminating technique due to A. Varga [8] to solve equation (A 1 ), i.e. we can decide algorithmically the problem of the existence of non-trivial solutions.…”

Section: The Case Of Algebraic Parametersmentioning

confidence: 99%

Nontrivial solutions of linear functional equations: methods and examples

Varga¹,

Vincze²

2015

OpMath

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“…Another interesting pure case when all the inner (or the outer) parameters are algebraic numbers over the rationals. Then we have an eliminating technic due to A. Varga [2] to solve equation (3). The mixed cases (algebraically dependent systems of parameters containing transcendental numbers) are open problems.…”

Section: Theorem 13 [4]mentioning

confidence: 99%

“…Recall that if all the outer parameters are algebraic numbers then we have an alternative method to check the existence of the non-zero additive solution by an algorithm due to A. Varga [2]. II.…”

Section: The Existence Of a Nontrivialmentioning

confidence: 99%

Algebraic methods for the solution of linear functional equations

2015

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belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the most earlier results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.

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“…In one of our main reference work [8] (see also [7]) the authors prove that all the elements of the hyperplane form algebraically dependent systems if and only if the coefficients α 1 , . .…”

Section: The Case Of Algebraically Independent Parametersmentioning

confidence: 99%

“…In this case we have a constructive method due to Varga [7] to solve functional Eq. (1.3) as follows: since all the inner parameters are algebraic, the extension of the rationals with elements β i 's can be written into the form Q(β 1 , .…”

Section: The Case Of Algebraic Parametersmentioning

confidence: 99%

On the characteristic polynomials of linear functional equations

Vincze

Varga

2015

Period Math Hung

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The solutions of a linear functional equation are typically generalized polynomials. The existence of their non-trivial monomial terms strongly depends on the algebraic properties of some related families of parameters. In extremal cases (the parameters are algebraic numbers or the parameters form an algebraically independent system) we have elegant methods to decide the existence of non-trivial solutions. In this paper we are going to extend and unify the treatment of the existence problem by introducing the characteristic polynomials of a linear functional equation such that the algebraic properties of the roots allows us to conclude the existence of non-trivial solutions.

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On additive solutions of a linear equation

Cited by 8 publications

References 3 publications

Nontrivial solutions of linear functional equations: methods and examples

Nontrivial solutions of linear functional equations: methods and examples

Algebraic methods for the solution of linear functional equations

On the characteristic polynomials of linear functional equations

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