next | previous | forward | backward | up | top | index | toc | Macaulay2 website
MixedMultiplicity :: secMilnorNumbers

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

Synopsis

Description

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 $e_0(m|J(f)),...,e_{d-1}(m|J(f))$, where $m$ is the maximal homogeneous ideal of $R$ and $J(f)$ is the Jacobian ideal of $f$.

i1 : k = frac(QQ[t])

o1 = k

o1 : FractionField
i2 : R = k[x,y,z]

o2 = R

o2 : PolynomialRing
i3 : secMilnorNumbers(z^5 + t*y^6*z + x*y^7 + x^15)

o3 = HashTable{0 => 1 }
               1 => 4
               2 => 26

o3 : HashTable
i4 : secMilnorNumbers(z^5 + x*y^7 + x^15)

o4 = HashTable{0 => 1 }
               1 => 4
               2 => 28

o4 : HashTable

Ways to use secMilnorNumbers :

For the programmer

The object secMilnorNumbers is a method function with options.