eulerObsMatrix -- Computes a matrix containing the strata-wise values of the local Euler obstruction function.

Below we find the value of the Euler obstruction function (also called the local Euler obstruction function) for all strata pairs. We begin by computing a Whitney stratification to input into the eulerObsMatrix function. We will use the Whitney cusp as our example variety, which we denote $X$.

Now we are ready to compute the Euler obstructions of strata.

The strata (closures) are represented as two element lists, the first element being the dimension and the second the defining equations. The order of this list is used in the matrix of Euler obstruction values.

First fix some notation. Suppose $V_i\subset V_j$ are two strata closures appearing in the list at positions $i$ and $j$, $i\leq j$. Let $Z$ be the open strata corresponding to $V_i$; $Z$ consists of all points in $V_i$ which are not in any strata closure of dimension strictly less than that of $V_i$.

With this notation in hand, the entry in row i and column j of the matrix Eu is $Eu_{V_j}(Z)$, which is the value of the local Euler obstruction function associated to $V_j$, $Eu_{V_j}: V_j\to \mathbb{Z}$, at any point in $Z$.

For more discussion see Remark 4.2 of reference [3] (Martin Helmer, Felix Tellander. "Spectral Decomposition of Euler-Mellin Integrals". Arxiv: 2505.12458.) Note that $Eu_{V_j}$ is defined to have value 0 outside $V_j$, meaning if some strata closure $V_l$ is not contained in $V_j$ the corresponding matrix row and column will be zero.

Note, this algorithm implementation contains steps which are probabilistic. To be confident of the answer it is advised to confirm by running twice.

Lets look at our specific example. Consider the last column of the matrix Eu. Since the last entry in our ordered list is the defining equation of the Whitney cusp $Y$, this column gives the the values of $Eu_Y$, the Euler obstruction function of $Y$.

The first entry of the column

is the value of $Eu_Y$ at any point in $Z=Y-V(x,y)=\{(x,y,z)|y^2+x^3-x^2*z^2=0, \; x\neq 0,\;y\neq 0\} $.

The second entry in of the column

is the value of $Eu_Y$ at any point in $Z=V(x,y)-V(x,y,z)=\{(x,y,z)|x=y=0, \; z\neq 0\} $. And the final entry of the column is the value of $Eu_Y$ at the point $(0,0,0)$.

There are also several different options to preform the underlying polar variety calculations. The default algorithm uses the M2 saturate command to compute the polar variteies, this option is Algorithm=>. The other options are: Algorithm=>"msolve" and Algorithm=>"M2F4". The Algorithm=>"msolve" only works in versions 1.25.06 and above of Macualay2. The Algorithm=>"M2F4" is mostly beneficial when working over a finite field. Note that over a finite field we can still sometimes obtain useful information about the stratification, but the coefficients appearing in the resulting polynomials may (or likely will) be incorrect.