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# vFunction -- compute the $\text{v}$-function of monomial ideal $I$

## Synopsis

• Usage:
vFunction I
• Inputs:
• I, an ideal, a monomial ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
• Optional inputs:
• control => an integer, default value 2, can be changed to increase reliability
• Outputs:
• l, a list, the first element represents the slope of the line, while the second represents the $y$-intercept

## Description

For $k\gg0$, the numerical function $k\mapsto\text{v}(I^k)$ is a linear function of the form $mk+q$, with $m=\alpha(I)$ the initial degree of $I$ and $q\ge-1$ an integer. This method computes the pair $(\alpha(I),q)$.

 i1 : S = QQ[x_1..x_3]; i2 : I = ideal(x_1*x_2,x_1*x_3,x_2*x_3); o2 : Ideal of S i3 : vFunction I o3 = {2, -1} o3 : List i4 : vFunction(I,control=>3) o4 = {2, -1} o4 : List
 i5 : S = QQ[x_1..x_3]; i6 : I = ideal(x_1,x_2)*ideal(x_1,x_3)*ideal(x_2,x_3); o6 : Ideal of S i7 : vFunction I o7 = {3, -1} o7 : List

• vNumber -- compute the $\text{v}$-number of homogeneous ideal $I$
• vNumberP -- compute the $\text{v}_\mathfrak{p}$-number of homogeneous ideal $I$
• vFunctionP -- compute the $\text{v}_\mathfrak{p}$-function of monomial ideal $I$

## Ways to use vFunction :

• vFunction(Ideal)

## For the programmer

The object vFunction is .