# waldschmidt -- computes the Waldschmidt constant for a homogeneous ideal.

## Synopsis

• Usage:
waldschmidt(I)
• Inputs:
• Optional inputs:
• SampleSize => ..., default value 5, optional parameter used for approximating asymptotic invariants that are defined as limits.
• Outputs:

## Description

The Waldschmidt constant for a homogeneous ideal I is defined as $waldschmidt(I)=lim_{n\to\infty} \frac{\alpha(I^{(n)})}{n}$, where $\alpha(J)$ denotes the smallest degree of a nonzero element in a given homogeneous ideal $J$. The limit of the sequence $\frac{\alpha(I^{(n)})}{n}$ exists because of the subadditivity of $\alpha$ and is equal to the infimum of the sequence $\frac{\alpha(I^{(n)})}{n}$.

The Waldschmidt constant can be computed for monomial ideals as the smallest value of the sum of the coordinates over all the points of the symbolic polyhedron. The function uses this method to return an exact answer for the Waldschmidt constant of a monomial ideal.

For ideals that are not monomial, we give an approximation of the Waldschmidt constant by taking the minimum value of $\frac{\alpha(I^{(n)})}{n}$ over a finite number of exponents $n$, namely for $n$ from 1 to the optional parameter SampleSize.

 i1 : R = QQ[x,y,z]; i2 : I = ideal(x*y,y*z,x*z); o2 : Ideal of R i3 : waldschmidt(I) 3 o3 = - 2 o3 : QQ
 i4 : R = QQ[x,y,z]; i5 : J = ideal (x*(y^3-z^3),y*(z^3-x^3),z*(x^3-y^3)); o5 : Ideal of R i6 : waldschmidt(J, SampleSize=>5) o6 = 3 o6 : QQ