Numerical Primary Decomposition

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(a) Write a Macaulay2 function that, given $I\subset k[x_1,\ldots,x_n]$, constructs a deflation ideal $I^{(1)} \subset k[x_1,\ldots,x_n,\lambda_1,\ldots \lambda_n]$. (You may set $k=\bQ$ or let $k$ be one of the input parameters of your function.)

(b) Consider

(1)
\begin{align} I = \<x^2(y+1),xy(y+1)\>. \end{align}

Verify that the deflated variety $X^{(1)}$ has 4 components; find the associated primes of $I^{(1)}$. Which component corresponds to a {\em pseudo-component} of $I$?

(b) Write a Macaulay2 function to construct $I^{(2)}$.

(c) Find the (symbolic) primary decomposition of $I^{(2)}$ and identify 4 prime components that correspond to the components and a pseudo-component of $I$.