secMilnorNumbers -- Compute the sectional Milnor numbers of a hypersurface with an isolated singularity

Let origin be an isolated singular point of a complex analytic hypersurface $H = V(f).$ The $\mathbb{C}$-vector space dimension of $\mathbb{C}\{x_0,\ldots,x_n\}/(f_{x_0},\ldots,f_{x_n})$ is called the Milnor number of the hypersurface $H$ at the origin. Let $(X, 0)$ be a germ of a hypersurface in $\mathbb{C}^{n+1}$ with an isolated singularity at the origin. The Milnor number of $X \cap E$, where $E$ is a general linear subspace of dimension $i$ passing through the origin, is called the $i$-th sectional Milnor number of $X$. B. Teissier identified the $i$-th sectional Milnor number with the $i$-th mixed multiplicity of the maximal homogeneous ideal of the polynomial ring and the Jacobian ideal of $f.$

Let $f$ be an element of a polynomial ring $R$ with characteristic zero, and let $d$ be the dimension of $R$. The function computes the sectional Milnor numbers by computing the mixed multiplicities $e_0(m|J(f)),\ldots,e_{d}(m|J(f))$, where $m$ is the maximal homogeneous ideal of $R$ and $J(f)$ is the Jacobian ideal of $f$ of height $d$. Note that in this case, the last sectional Milnor number $e_d(m|J(f))$ is the Milnor number of $f.$

In this example, the 2-sectional Milnor number, which is the Milnor number of a general hypersurface section, is 28. The Milnor number, which is the last sectional Milnor number, is 364.

The user is supposed to check that the given polynomial defines an isolated singularity at the homogeneous maximal ideal.