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# vNumberP -- compute the $\text{v}_\mathfrak{p}$-number of homogeneous ideal $I$

## Synopsis

• Usage:
vNumberP(I,p)
• Inputs:
• I, an ideal, a homogeneous ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
• p, an ideal, a associated prime of $I$
• Outputs:

## Description

This method computes the $\text{v}_{\mathfrak{p}}$-number of $I$: $$\text{v}_{\mathfrak{p}}(I)=\min\{\text{deg}(f):f\in S\ \text{is homogeneous, and}\ (I:f)=\mathfrak{p}\}.$$

 i1 : S = QQ[x_1..x_3]; i2 : I = ideal(x_1*x_2,x_1*x_3,x_2*x_3) o2 = ideal (x x , x x , x x ) 1 2 1 3 2 3 o2 : Ideal of S i3 : p = ideal(x_1,x_2) o3 = ideal (x , x ) 1 2 o3 : Ideal of S i4 : vNumberP(I,p) o4 = 1
 i5 : S = QQ[x_1..x_4]; i6 : I = ideal(x_1^3*x_2+x_3^4,x_1+x_2+x_4,x_3^3); o6 : Ideal of S i7 : p = (ass I)#0; o7 : Ideal of S i8 : vNumberP(I,p) o8 = 5

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