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In their paper Boulier et al. (2009) described the Rosenfeld-Grobner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of polynomial differential equations, ordinary or with partial derivatives. In order to enhance the efficiency of this algorithm, they proposed their analog of Buchberger's criteria to avoid useless reductions to zero. For example, they showed that if $p$ and $q$ are two differential polynomials which are linear, homogeneous, in one differential indeterminate, with constant coefficients and with leaders $\theta u$ and $\phi u$, respectively so that $\theta$ and $\phi$ are disjoint then the delta-polynomial of $p$ and $q$ reduces to zero w.r.t. the set $\{p,q\}$. In this paper we generalize this result showing that it remains true if $p$ and $q$ are products of differential polynomials which are linear, homogeneous, in the same differential indeterminate, with constant coefficients and $\theta$ and $\phi$ are disjoint where $\theta u$ and $\phi u$ are leaders of $p$ and $q$, respectively. We have implemented the Rosenfeld-Grobner algorithm and our refined version on the same platform in Maple and compare them via a set of benchmarks.
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