Application of the p-Adic Topology on ℤ to the Problem of Finding Solutions in Integers of an Implicit Linear Difference Equation

Gerasimov, Vladislav; Gefter, S. L.; Goncharuk, A. B.

doi:10.1007/s10958-018-4072-x

Cited by 8 publications

(5 citation statements)

References 3 publications

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“…Since r < 1, then |y n | tends to 0, so the series ( 5) converges. To verify that ( 5) satisfies (1), let us substitute it into the equation. Taking into account that {y n } ∞ n=0 satisfies (4), we obtain…”

Section: Existence and Uniqueness Theoremsmentioning

confidence: 99%

“…In the article [1] a simple interesting fact about recurrence equations is discovered: there is shown that the infinite implicit system of linear equations in variables x 0 , x 1 , x 2 , . .…”

Section: Introductionmentioning

confidence: 99%

“…It is proven that there exists a unique solution from the ring of p-adic integers Z p and this solution is found explicitly as a sum of series converging in Z p with respect to the well-known non-Archimedean valuation (see [1,Cor.2.1]).…”

Section: Introductionmentioning

confidence: 99%

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Implicit linear difference equations over a non-Archi-medean ring

2021

MAMM

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Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering p-adic integers ~ the completion of the ring of integers with respect to the non-Archimedean p-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of p-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of p-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.

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Section: Existence and Uniqueness Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Implicit linear difference equations over a non-Archi-medean ring

2021

MAMM

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“…Therefore, it appears to be interesting to investigate the problem of solving an implicit linear difference equation with non-invertible coefficients from an arbitrary commutative ring. Recently, implicit difference equations over integral domains were studied in [7], and more detailed over the ring of integers in [8]- [10].…”

Section: Introductionmentioning

confidence: 99%

Implicit linear difference equation over residue class rings

Heneralov,

Piven'

2024

ADM

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“…Therefore the interesting problem of the investigation of the implicit linear difference equation with non-invertible coefficients from the any commutative ring appeared. At the moment implicit difference equations over the ring of integers were studied more detailed [7]- [9]. In [10] these equations in different classes of topological vector spaces were investigated.…”

Section: Introductionmentioning

confidence: 99%