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MixedMultiplicity :: secMilnorNumbers

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 $(X, x)$ be a germ of a hypersurface in $\mathbb{C}^{n+1}$ with an isolated singularity at $x$. The Milnor number of $X \cap E$, where $E$ is a general linear subspace of dimension $i$ passing through $x,$ 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 $R$ and let $d$ be the dimension of $R$. The function computes the first $d-1$ sectional Milnor numbers by computing the mixed multiplicities $e_0(m|J(f)),\ldots,e_{d-1}(m|J(f))$, where $m$ is the maximal homogeneous ideal of $R$ and $J(f)$ is the Jacobian ideal of $f$.

i1 : R = QQ[x,y,z];
i2 : secMilnorNumbers(z^5 + x*y^7 + x^15)

o2 = HashTable{0 => 1 }
               1 => 4
               2 => 28

o2 : HashTable
i3 : secMilnorNumbers(z^3 + y^3 + x^3)

o3 = HashTable{0 => 1}
               1 => 2
               2 => 4

o3 : HashTable

Ways to use secMilnorNumbers :

For the programmer

The object secMilnorNumbers is a method function.