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Linear differential elimination for analytic functions. (English) Zbl 1230.12005

Let \(\Omega\) be a connected open subset of complex \(n\) space. Let \(K\) be the field of meromorphic functions on \(\Omega\) and let \(R\) be the (skew) polynomial ring in the \(n\) partial derivatives on \(n\) space. Let \(g=(g_1, \dots, g_k)\) be a tuple of non-zero analytic complex functions on \(\Omega\). Assume that for each \(i\), \(1 \leq i \leq k\) there is is a \(\nu(i) <n\) with \(\nu(i)\) functionally independent complex analytic functions \(\alpha_{i,j}\) such that the Jacobian of the function \(\alpha_i\) to complex \(\nu(i)\) space determined by them has rank \(\nu(i)\) everywhere on \(\Omega\). Let \(\alpha=(\alpha_1, \dots, \alpha_k)\). Then an analytic complex valued \(u\) function on \(\Omega\) is called \((\alpha, g)\) representable if there are complex functions \(f_i\) on \(\alpha_i(\Omega)\) such that the compositions \(f_i\circ \alpha_i\) are analytic and \(u(z)=\sum f_i(\alpha_i(z))g_i(z)\) for all \(z \in \Omega\). If there is a dense open subset of points \(p\) of \(\Omega\) such that there is an open neighborhood \(\Omega^\prime\) of \(p\) such that \(u\) restricted to \(\Omega^\prime\) is \((\alpha^\prime, g^\prime)\) representable, where the primes denote restriction to \(\Omega^\prime\), then \(u\) is essentially \((\alpha, g)\) representable. The authors show that there is an ideal \(I(\alpha, g)\) of \(R\) such that \(u\) is essentially \((\alpha, g)\) representable if and only if it is annihilated by \(I(\alpha, g)\). The authors provide an algorithm which, for each \(d\) computes a Janet basis of the order \(d\) or smaller elements of \(I(\alpha, g)\). They illustrate in examples and applications how to use this to compute representations for various choices of \(\alpha\), \(g\), and \(u\).

MSC:

12H05 Differential algebra
13N10 Commutative rings of differential operators and their modules
15A03 Vector spaces, linear dependence, rank, lineability
32A17 Special families of functions of several complex variables
68W30 Symbolic computation and algebraic computation

Software:

CRACK; Janet
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Full Text: DOI

References:

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