# differentialPrimaryDecomposition -- compute a differential primary decomposition

## Synopsis

• Usage:
differentialPrimaryDecomposition U
• Inputs:
• U, , a submodule of a free module, or an ideal
• Outputs:
• a list, a minimal differential primary decomposition of U

## Description

Let $R$ be a polynomial ring over a field $K$. Given a submodule $U$ of an $R$-module $M$, a differential primary decomposition of $U$ in $M$ is a list of pairs $(p_1, A_1), ..., (p_k, A_k)$ where $p_1, ..., p_k$ are the associated primes of $M/U$ and $A_i \subseteq \operatorname{Diff}_{R/K}(M, R/p_i)$ are differential operators satisfying $$U_p = \bigcap_{p_i \subseteq p} \{ w \in M_p : \delta(w) = 0 , \ \forall \delta \in A_i \}.$$ This notion was introduced in [2] (cf. Definition 4.1), in which it was shown that the size of a differential primary decomposition (which is defined to be $\sum_{i=1}^k |A_i|$) is at least amult(U), and moreover differential primary decompositions of size equal to amult(U) exist (and are called minimal).

This method contains an implementation of Algorithm 4.6 in [2].

The following example appears as Example 6.2 in [1]:

 i1 : R = QQ[x_1,x_2,x_3] o1 = R o1 : PolynomialRing i2 : U = image matrix {{x_1^2,x_1*x_2,x_1*x_3}, {x_2^2,x_2*x_3,x_3^2}} o2 = image | x_1^2 x_1x_2 x_1x_3 | | x_2^2 x_2x_3 x_3^2 | 2 o2 : R-module, submodule of R i3 : differentialPrimaryDecomposition U 2 o3 = {{ideal x , {| 1 |}}, {ideal(x - x x ), {| -x_3 |}}, {ideal (x , x ), 1 | 0 | 2 1 3 | x_1 | 3 2 ------------------------------------------------------------------------ {| 0 |}}} | dx_3 | o3 : List