Delta- and Quasi-Regularity for Polynomial Ideals
Co-author(s): Marcus Hausdorf and Mehdi Sahbi
Reference: Global Integrability of Field Theories - GIFT 2006, J. Calmet, W.M. Seiler, R.W. Tucker (eds.), Universitätsverlag Karlsruhe 2006, pp. 179-200

This article treats three problems: (i) the effective determination of delta-regular coordinates for a polynomial ideal, (ii) implications of the existence of a Pommaret basis for a monomial ideal, and (iii) the relation between delta-regularity and quasi-regularity in the sense of Serre.

Concerning (i) we recall the results of [22] for differential equations in the polynomial version of [28] . Concerning (ii) we introduce the notion of a quasi-stable ideal as a monomial ideal possessing a Pommaret ideal. Then we show that this concept coincides with monomial ideals of nested type recently introduced by Bermejo and Gimenez in their work on the Castelnuovo-Mumford regularity. This provides at once a number of algebraic characterisations of such ideals. One of these characterisations also provides a simple effective criterion for delta-regularity via colon ideals. We show that it is closely related to the criterion discussed under (i). As a by-product we obtain a simple description of an irredundant primary decomposition of a quasi-stable ideal.

Serre provided a dual version of the Cartan test for involution formulated for polynomial modules. Like all versions of Cartan's test it is coordinate dependent and consequently Serre introduced the notion of a quasi-regular sequence (generalising the classical concept of a regular sequence). The test requires the existence of a quasi-regular basis. We show now that quasi-regularity and delta-regularity are equivalent in the sense that coordinates are delta-regular for an ideal I, if and only if they are quasi-regular for the factor algebra P/I.

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