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Sur la calculabilité des déterminants de matrices d’opérateurs différentiels. (On the calculability of determinants of matrices of differential operators). (French) Zbl 0647.16003
This paper addresses the calculability (by means of finite algorithms) of certain determinant functions on matrices over the quotient division ring K of an Ore domain D. Given a suitable monoid homomorphism \(\phi\) from D to a commutative monoid M, a function \(\det_{\phi}\) is constructed by composing the usual Dieudonné determinant (with values in the abelianization of \(K^*)\) with the map induced by \(\phi\) from \((K^*)^{ab}\) to a localization of M. Suitable calculability hypotheses on D, M, \(\phi\) imply that \(\det_{\phi}\) is calculable. This is specialized to certain formal differential operator rings D over commutative domains C with several commuting derivations. In the most general case, M is a monoid of homogeneous polynomials and \(\phi\) assigns to each differential operator its principal part. In another case, C is a rational function field or a Laurent series field, and \(\phi\) assigns “formal Newton polyhedra” (analogous to the classical Newton polyhedra of polynomials) to differential operators. Algorithms for the relevant calculable quantities are given, and some are worked out in the formal calculus MACSYMA.
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16-04 Software, source code, etc. for problems pertaining to associative rings and algebras
15A15 Determinants, permanents, traces, other special matrix functions