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

## Synopsis

• Usage:
secMilnorNumbers f
• Inputs:
• f, , polynomial
• Optional inputs:
• BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
• DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step.
• MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
• PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
• Strategy => ..., default value null, Choose a strategy for the saturation step
• Outputs:
• , First $d-1$ sectional Milnor numbers, where $d$ is the dimension of the polynomial ring

## 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 :

• secMilnorNumbers(RingElement) (missing documentation)

## For the programmer

The object secMilnorNumbers is .