Let {wi,j}1≤i≤n,1≤j≤s ⊂ Lm = F (X1, . . . , Xm)[ ∂ ∂X1 , . . . , ∂ ∂Xm ] be linear partial differential operators of orders with respect to ∂ ∂X1 , . . . , ∂ ∂Xm at most d. We prove an upper bound n(4mdmin{n, s}) (2(m−t)) on the leading coefficient of the Hilbert-Kolchin polynomial of the left Lm-module 〈{w1,j , . . . , wn,j}1≤j≤s〉 ⊂ L n m having the differential type t (also being equal to the...

The multi-homogeneous Bézout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the Bézout number. In this paper, it is proved that the problem of computing, or even estima...

Let [Formula: see text] represent a set of proper nonzero ideals (respectively, t-ideals text]) an integral domain and let P be valid property D. We say S(D) meets (denoted if each is contained in ideal satisfying P. If cannot controlled. When does not imply I(R) while implies usually. with twist text]written such that, for some study as its predecessor. A modification the above approach used t...

in this paper, we study some ring theoretic properties of the amalgamated duplication ring $rbowtie i$ of a commutative noetherian ring $r$ along an ideal $i$ of $r$ which was introduced by d'anna and fontana. indeed, it is determined that when $rbowtie i$ satisfies serre's conditions $(r_n)$ and $(s_n)$, and when is a normal ring, a generalized cohen-macaulay ring and finally a filter ring.