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# vFunctionP -- compute the $\text{v}_\mathfrak{p}$-function of monomial ideal $I$

## Synopsis

• Usage:
vFunctionP(I,p)
• Inputs:
• I, an ideal, a monomial ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
• p, an ideal, a stable prime of $I$
• 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 $k\mapsto\text{v}_\mathfrak{p}(I^k)$ is a linear function of the form $mk+q$, with $m$ and $q$ integers. This method computes the pair $(m,q)$.

 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 : p = ideal(x_1,x_2); o3 : Ideal of S i4 : vFunctionP(I,p) 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 : p = ideal(x_1,x_2,x_3); o7 : Ideal of S i8 : vFunctionP(I,p,control=>3) o8 = {3, -1} o8 : List

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