Über das macaulaysche inverse system und dessen bedeutung für die theorie der linearen differentialgleichungen mit konstanten koeffizienten

Gröbner, Wolfgang

doi:10.1007/bf02948939

Cited by 17 publications

(20 citation statements)

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“…Inexplicitly, he used the ring isomorphism from the ring of polynomials in several variables with coefficients in an arbitrary field K to the ring of differential operators with constant coefficients. Note that, this isomorphism was proved explicitly more than fifteen years later by W. Gröbner in [21] in a modern algebraic language as follows:…”

Section: Monomial Partial Differential Equation Systemsmentioning

confidence: 88%

Maurice Janet’s algorithms on systems of linear partial differential equations

Iohara

Malbos

2020

Arch. Hist. Exact Sci.

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This article presents the emergence of formal methods in theory of partial differential equations (PDE) in the french school of mathematics through Janet's work in the period 1913-1930. In his thesis and in a series of articles published during this period, M. Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the unicity of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. M. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the 20th century in various algebraic contexts.

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Section: Monomial Partial Differential Equation Systemsmentioning

confidence: 88%

Maurice Janet’s algorithms on systems of linear partial differential equations

Iohara

Malbos

2020

Arch. Hist. Exact Sci.

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“…Theorem 2.17 can be reformulated in terms of solutions of partial differential equations, using the relation between Artinian algebras and polynomial-exponentials PolExp. This duality between polynomial equations and partial differential equations with constant coefficients goes back to [61] and has been further studied and extended for instance in [33], [25], [53], [52], [35]. In the case of a non-Artinian algebra, the solutions on an open convex domain are in the closure of the set of polynomial-exponential solutions (see e.g.…”

Section: Artinian Algebramentioning

confidence: 96%

Polynomial–Exponential Decomposition From Moments

Mourrain

2017

Found Comput Math

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We analyze the decomposition problem of multivariate polynomial-exponential functions from their truncated series and present new algorithms to compute their decomposition.Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial-exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the isolated points of the characteristic variety.Using the properties of Hankel operators, we establish a correspondence between polynomial-exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial-exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol.A generalization of Prony's approach to multivariate decomposition problems is presented, exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial-exponential decomposition, using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Carathéodory-Fejér decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram-Schmidt orthogonalization process and to decompose polynomial-exponential series.A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial-exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial-exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.

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“…If we consider ν : Π → Π , p → ν(p), as a mapping, its image, the linear space N := ν(Π ) ⊂ Π is a canonical interpolation space. With the specific canonical choice (5) of the inner product, the normal form space is the Macaulay inverse system, as it was named in [14,15].…”

Section: Theorem 1 ([36]) If H Is An H-basis For H Then the Remaindermentioning

confidence: 99%