This is VNumber.info, produced by Macaulay2, version 1.25.11

VNumber -- a package for computing the v-number and v-function
**************************************************************

Description
===========

VNumber is a package for computing the v-number of homogeneous ideals and
v-function of monomial ideals

Authors
=======

  * Antonino Ficarra (see https://www.researchgate.net/profile/Antonino-Ficarra
    ) <antficarra@unime.it>
  * Emanuele Sgroi (see https://www.researchgate.net/profile/Emanuele-Sgroi )
    <emasgroi@unime.it>

Version
=======

This documentation describes version 1.0 of VNumber, released September 12,
2024.

Citation
========

If you have used this package in your research, please cite it as follows:

+-----------------------------------------------------------------------------+
|@misc{VNumberSource,                                                         |
|  title = {{VNumber: compute v-number of homogeneous ideals and v-function of|
|  author = {Antonino Ficarra and Emanuele Sgroi},                            |
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} |
|}                                                                            |
|-----------------------------------------------------------------------------|
|monomial ideals. Version~1.0}},                                              |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * isStablePrime -- test whether a prime ideal is a
        stable prime of a ideal $I$
      * reesMap -- compute the Rees map of $I$
      * soc -- compute the $\text{Soc}_\mathfrak{p}^*(I)$, where
        $I$ is a monomial ideal
      * stableMax -- compute the set of stable primes of a
        monomial ideal that are maximal with respect to the inclusion.
      * stablePrimes -- compute the set of stable primes
        of a monomial ideal
      * vFunction -- compute the $\text{v}$-function of
        monomial ideal $I$
      * vFunctionP -- compute the
        $\text{v}_\mathfrak{p}$-function of monomial ideal $I$
      * vNumber -- compute the $\text{v}$-number of homogeneous
        ideal $I$
      * vNumberP -- compute the $\text{v}_\mathfrak{p}$-number
        of homogeneous ideal $I$
  * Methods
      * "isStablePrime(Ideal,Ideal)" -- see isStablePrime,
        -- test whether a prime ideal is a stable prime of a ideal $I$
      * "reesMap(Ideal)" -- see reesMap, -- compute the Rees map
        of $I$
      * "soc(Ideal,Ideal)" -- see soc, -- compute the
        $\text{Soc}_\mathfrak{p}^*(I)$, where $I$ is a monomial ideal
      * "stableMax(Ideal)" -- see stableMax, -- compute the
        set of stable primes of a monomial ideal that are maximal with respect
        to the inclusion.
      * "stablePrimes(Ideal)" -- see stablePrimes, --
        compute the set of stable primes of a monomial ideal
      * "vFunction(Ideal)" -- see vFunction, -- compute the
        $\text{v}$-function of monomial ideal $I$
      * "vFunctionP(Ideal,Ideal)" -- see vFunctionP, --
        compute the $\text{v}_\mathfrak{p}$-function of monomial ideal $I$
      * "vNumber(Ideal)" -- see vNumber, -- compute the
        $\text{v}$-number of homogeneous ideal $I$
      * "vNumberP(Ideal,Ideal)" -- see vNumberP, -- compute the
        $\text{v}_\mathfrak{p}$-number of homogeneous ideal $I$
  * Symbols
      * control -- optional argument for vFunctionP and
        vFunction

For the programmer
==================

The object VNumber, is a package (Macaulay2Doc),
defined in VNumber.m2.

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

control -- optional argument for vFunctionP and vFunction
*********************************************************

Description
===========

By default this argument is set to 2. Whether it is equal to 2 the vFunctionP
and vFunction methods terminate after finding three "aligned" values
$\text{v}_\mathfrak{p}(I^k),\text{v}_\mathfrak{p}(I^{k+1}),\text{v}_\mathfrak{p
}(I^{k+2})$ of the $\text{v}_\mathfrak{p}$-function. That means that
$\text{v}_\mathfrak{p}(I^{k+1})-\text{v}_\mathfrak{p}(I^k)=\text{v}_\mathfrak{p
}(I^{k+2})-\text{v}_\mathfrak{p}(I^{k+1})$. By increasing its value, the
methods terminate after finding $\texttt{control}+1$ aligned values.

See also
========

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

Functions with optional argument named control:
===============================================

  * "vFunction(...,control=>...)"
  * "vFunctionP(...,control=>...)"

For the programmer
==================

The object control, is a symbol (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

isStablePrime -- test whether a prime ideal is a stable prime of a ideal $I$
****************************************************************************

  * Usage:
        isStablePrime(I,p)
  * Inputs:
      * I, an ideal (Macaulay2Doc), a monomial ideal of polynomial
        ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
      * p, an ideal (Macaulay2Doc), a prime monomial ideal of $S$
  * Outputs:
      * b, a Boolean value (Macaulay2Doc), whether
        $\mathfrak{p}$ is a stable prime of $I$

Description
===========

This method returns true if $\mathfrak{p}\in\text{Ass}^\infty(I)$, otherwise
false.

+---------------------------------------+
|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 : isStablePrime(I,p)                |
|                                       |
|o4 = true                              |
+---------------------------------------+
+------------------------------------+
|i5 : S = QQ[a..e];                  |
+------------------------------------+
|i6 : I = ideal(a*d,a*e,b*d,c*d,c*e);|
|                                    |
|o6 : Ideal of S                     |
+------------------------------------+
|i7 : p = ideal(a,c);                |
|                                    |
|o7 : Ideal of S                     |
+------------------------------------+
|i8 : isStablePrime(I,p)             |
|                                    |
|o8 = false                          |
+------------------------------------+

See also
========

  * stablePrimes -- compute the set of stable primes of a
    monomial ideal
  * stableMax -- compute the set of stable primes of a
    monomial ideal that are maximal with respect to the inclusion.

Ways to use isStablePrime:
==========================

  * "isStablePrime(Ideal,Ideal)"

For the programmer
==================

The object isStablePrime, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

reesMap -- compute the Rees map of $I$
**************************************

  * Usage:
        reesMap I
  * Inputs:
      * I, an ideal (Macaulay2Doc), a homogeneous ideal of
        polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
  * Outputs:
      * g, a ring map (Macaulay2Doc), the Rees map of $I$

Description
===========

Given a homogeneous ideal $I$, with minimal homogeneous generating set
$\{f_1,\dots,f_m\}$, this method yields the bigraded map $g:
\mathbb{K}[x_1,\dots,x_n,y_1,\dots,y_m] \to \mathbb{K}[x_1,\dots,x_n,t]$
defined by setting $g(x_i) = x_i$ for $1 \le i \le n$ and $g(y_j) = f_j t$ for
$1 \le j \le m$ and with $\text{bideg}(x_i) = (1,0)$ and $\text{bideg}(y_j) =
(\text{deg}(f_j), 1)$.

+-----------------------------------------------------------------------------+
|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 : reesMap I                                                               |
|                                                                             |
|o3 = map (QQ[x ..x , t], QQ[x ..x , y ..y ], {x , x , x , x x t, x x t,      |
|              1   3          1   3   1   3     1   2   3   1 2    1 3        |
|                                                                             |
|o3 : RingMap QQ[x ..x , t] <-- QQ[x ..x , y ..y ]                            |
|                 1   3             1   3   1   3                             |
|-----------------------------------------------------------------------------|
|x x t})                                                                      |
| 2 3                                                                         |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i4 : S = QQ[x_1..x_4];                                                       |
+-----------------------------------------------------------------------------+
|i5 : I = ideal(x_1^3*x_2+x_3^4,x_1+x_2+x_4,x_3^3);                           |
|                                                                             |
|o5 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i6 : reesMap I                                                               |
|                                                                             |
|                                                               3       4     |
|o6 = map (QQ[x ..x , t], QQ[x ..x , y ..y ], {x , x , x , x , x x t + x t,   |
|              1   4          1   4   1   3     1   2   3   4   1 2     3     |
|                                                                             |
|o6 : RingMap QQ[x ..x , t] <-- QQ[x ..x , y ..y ]                            |
|                 1   4             1   4   1   3                             |
|-----------------------------------------------------------------------------|
|                  3                                                          |
|x t + x t + x t, x t})                                                       |
| 1     2     4    3                                                          |
+-----------------------------------------------------------------------------+

Ways to use reesMap:
====================

  * "reesMap(Ideal)"

For the programmer
==================

The object reesMap, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

soc -- compute the $\text{Soc}_\mathfrak{p}^*(I)$, where $I$ is a monomial ideal
********************************************************************************

  * Usage:
        soc(I,p)
  * Inputs:
      * I, an ideal (Macaulay2Doc), a monomial ideal of polynomial
        ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
      * p, an ideal (Macaulay2Doc), a stable prime of $I$
  * Outputs:
      * M, a module (Macaulay2Doc),

Description
===========

This method computes the module $\text{Soc}_\mathfrak{p}^*(I)$. Let
$\mathcal{R}(I)=\bigoplus_{k\ge0}I^k$ be the Rees algebra of $I$. For the prime
$\mathfrak{p}\in\text{Ass}^\infty(I)$, we set
$$X_\mathfrak{p}=\begin{cases}S&\text{if}\
\mathfrak{p}\in\text{Max}^\infty(I),\\
\prod\{\mathfrak{q}\in\text{Ass}^\infty(I):\mathfrak{p}\subsetneq\mathfrak{q
}\}&\text{otherwise}.\end{cases}$$ Let $\mathcal{R}'=\bigoplus_{k\ge0}I^{k+1}$.
We set $$\text{Soc}_\mathfrak{p}^*(I)=\frac{(\mathcal{R}':_{\mathcal{R}(I)}\
mathfrak{p}\mathcal{R}(I))}{(\mathcal{R}':_{\mathcal{R}(I)}(\mathfrak{p}+X_\
mathfrak{p}^\infty)\mathcal{R}(I))}.$$ Here
$X_\mathfrak{p}^\infty=\bigcup_{k\ge0}(\mathfrak{p}:\mathfrak{m}^k)$ is the
saturation of $\mathfrak{p}$ with respect to the maximal ideal
$\mathfrak{m}=(x_1,\dots,x_n)$. As proved by Conca, for all $k\gg0$ we have
$$\text{Soc}_\mathfrak{p}^*(I)_{(*,k)}=\frac{(I^{k+1}:\mathfrak{p})}{I^{k+1}:(\
mathfrak{p}+X_\mathfrak{p}^\infty)},$$ and the initial degree of this module is
the $\text{v}_\mathfrak{p}$-number of $I^{k+1}$.

+-----------------------------------------------------------------------------+
|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 : soc(I,p)                                                                |
|                                                                             |
|o4 = subquotient (| x_3 x_1x_2 |, | x_2x_3 x_1x_3 x_1x_2 x_                  |
|                                                                             |
|           QQ[x ..x , y ..y ]                                                |
|               1   3   1   3                                                 |
|o4 : -------------------------------module, subquotient of                   |
|     (- x y  + x y , - x y  + x y )                                          |
|         1 3    2 2     1 3    3 1                                           |
|-----------------------------------------------------------------------------|
|3y_1 |)                                                                      |
|                                                                             |
|/      QQ[x ..x , y ..y ]      \                                             |
||          1   3   1   3       |1                                            |
||------------------------------|                                             |
||(- x y  + x y , - x y  + x y )|                                             |
|\    1 3    2 2     1 3    3 1 /                                             |
+-----------------------------------------------------------------------------+

See also
========

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

Ways to use soc:
================

  * "soc(Ideal,Ideal)"

For the programmer
==================

The object soc, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

stableMax -- compute the set of stable primes of a monomial ideal that are maximal with respect to the inclusion.
*****************************************************************************************************************

  * Usage:
        stableMax I
  * Inputs:
      * I, an ideal (Macaulay2Doc), a monomial ideal of polynomial
        ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
  * Outputs:
      * l, a list (Macaulay2Doc), all maximal stable primes of $I$

Description
===========

This method computes $\text{Max}^\infty(I)$, the set of the stable primes of
$I$ that are maximal with respect to the inclusion.

+---------------------------------------+
|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 : stableMax I                       |
|                                       |
|o3 = {ideal (x , x , x )}              |
|              1   2   3                |
|                                       |
|o3 : List                              |
+---------------------------------------+
+-----------------------------------------------------+
|i4 : S = QQ[a..e];                                   |
+-----------------------------------------------------+
|i5 : I = ideal(a*d,a*e,b*d,c*d,c*e);                 |
|                                                     |
|o5 : Ideal of S                                      |
+-----------------------------------------------------+
|i6 : stableMax I                                     |
|                                                     |
|o6 = {ideal (c, b, a), ideal (d, c, a), ideal (e, d)}|
|                                                     |
|o6 : List                                            |
+-----------------------------------------------------+

See also
========

  * stablePrimes -- compute the set of stable primes of a
    monomial ideal

Ways to use stableMax:
======================

  * "stableMax(Ideal)"

For the programmer
==================

The object stableMax, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

stablePrimes -- compute the set of stable primes of a monomial ideal
********************************************************************

  * Usage:
        stablePrimes I
  * Inputs:
      * I, an ideal (Macaulay2Doc), a monomial ideal of polynomial
        ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
  * Outputs:
      * l, a list (Macaulay2Doc), all stable primes of $I$

Description
===========

By a classical result of Brodmann, $\text{Ass}(I^k)=\text{Ass}(I^{k+1})$ for
all $k\gg0$. We denote the common set $\text{Ass}(I^k)$ for $k\gg0$ by
$\text{Ass}^\infty(I)$. A prime ideal $\mathfrak{p}\subset S$ such that
$\mathfrak{p}\in \text{Ass}(I^k)$ for all $k \gg 0$ is called a stable prime of
$I$. This method computes $\text{Ass}^\infty(I)$, the set of the stable primes
of $I$.

+-------------------------------------------------------------------------+
|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 : stablePrimes I                                                      |
|                                                                         |
|o3 = {ideal (x , x ), ideal (x , x ), ideal (x , x ), ideal (x , x , x )}|
|              2   1           3   1           3   2           1   2   3  |
|                                                                         |
|o3 : List                                                                |
+-------------------------------------------------------------------------+
+-----------------------------------------------------+
|i4 : S = QQ[a..e];                                   |
+-----------------------------------------------------+
|i5 : I = ideal(a*d,a*e,b*d,c*d,c*e);                 |
|                                                     |
|o5 : Ideal of S                                      |
+-----------------------------------------------------+
|i6 : stablePrimes I                                  |
|                                                     |
|o6 = {ideal (c, b, a), ideal (d, c, a), ideal (e, d)}|
|                                                     |
|o6 : List                                            |
+-----------------------------------------------------+

See also
========

  * isStablePrime -- test whether a prime ideal is a
    stable prime of a ideal $I$
  * stableMax -- compute the set of stable primes of a
    monomial ideal that are maximal with respect to the inclusion.

Ways to use stablePrimes:
=========================

  * "stablePrimes(Ideal)"

For the programmer
==================

The object stablePrimes, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

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

  * Usage:
        vFunction I
  * Inputs:
      * I, an ideal (Macaulay2Doc), a monomial ideal of polynomial
        ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
  * Optional inputs (Macaulay2Doc),:
      * control, => an integer (Macaulay2Doc), default
        value 2, can be changed to increase reliability
  * Outputs:
      * l, a list (Macaulay2Doc), 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                                             |
+------------------------------------------------------+

See also
========

  * 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 a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

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

  * Usage:
        vFunctionP(I,p)
  * Inputs:
      * I, an ideal (Macaulay2Doc), a monomial ideal of polynomial
        ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
      * p, an ideal (Macaulay2Doc), a stable prime of $I$
  * Optional inputs (Macaulay2Doc),:
      * control, => an integer (Macaulay2Doc), default
        value 2, can be changed to increase reliability
  * Outputs:
      * l, a list (Macaulay2Doc), 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                                             |
+------------------------------------------------------+

See also
========

  * 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 $I$

Ways to use vFunctionP:
=======================

  * "vFunctionP(Ideal,Ideal)"

For the programmer
==================

The object vFunctionP, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

vNumber -- compute the $\text{v}$-number of homogeneous ideal $I$
*****************************************************************

  * Usage:
        vNumberP(I,p)
  * Inputs:
      * I, an ideal (Macaulay2Doc), a homogeneous ideal of
        polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
  * Outputs:
      * k, an integer (Macaulay2Doc),

Description
===========

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

+----------------------------------------+
|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 : vNumber(I)                         |
|                                        |
|o3 = 1                                  |
+----------------------------------------+
+--------------------------------------------------+
|i4 : S = QQ[x_1..x_4];                            |
+--------------------------------------------------+
|i5 : I = ideal(x_1^3*x_2+x_3^4,x_1+x_2+x_4,x_3^3);|
|                                                  |
|o5 : Ideal of S                                   |
+--------------------------------------------------+
|i6 : vNumber(I)                                   |
|                                                  |
|o6 = 5                                            |
+--------------------------------------------------+

See also
========

  * vNumberP -- compute the $\text{v}_\mathfrak{p}$-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 $I$

Ways to use vNumber:
====================

  * "vNumber(Ideal)"

For the programmer
==================

The object vNumber, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in VNumber.m2:547:0.

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

  * Usage:
        vNumberP(I,p)
  * Inputs:
      * I, an ideal (Macaulay2Doc), a homogeneous ideal of
        polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
      * p, an ideal (Macaulay2Doc), a associated prime of $I$
  * Outputs:
      * k, an integer (Macaulay2Doc),

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                                            |
+--------------------------------------------------+

See also
========

  * 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 $I$

Ways to use vNumberP:
=====================

  * "vNumberP(Ideal,Ideal)"

For the programmer
==================

The object vNumberP, is a method function (Macaulay2Doc).

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The source of this document is in VNumber.m2:547:0.