# Difference between revisions of "ApCoCoA-1:NCo.TruncatedGB"

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+ | {{Version|1}} | ||

<command> | <command> | ||

<title>NCo.TruncatedGB</title> | <title>NCo.TruncatedGB</title> | ||

<short_description> | <short_description> | ||

Compute a truncated Groebner basis of a finitely generated homogeneous two-sided ideal in a free monoid ring. | Compute a truncated Groebner basis of a finitely generated homogeneous two-sided ideal in a free monoid ring. | ||

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</short_description> | </short_description> | ||

<syntax> | <syntax> | ||

− | NCo.TruncatedGB(G:LIST, DB:INT | + | NCo.TruncatedGB(G:LIST, DB:INT):LIST |

</syntax> | </syntax> | ||

<description> | <description> | ||

+ | Given a word ordering and a homogeneous two-sided ideal <tt>I</tt>, a set of non-zero polynomials <tt>G</tt> is called a Groebner basis of <tt>I</tt> if the leading word set <tt>LW{G}</tt> generates the leading word ideal <tt>LW(I)</tt>. Note that it may not exist finite Groebner basis of the ideal <tt>I</tt>. Moreover, let <tt>D</tt> be a positive integer. Then the set <tt>{g in G | Deg(g)<=D}</tt> is a Groebner basis of the ideal <tt><f in I | Deg(f)<=D></tt> and is called a <em>D-truncated</em> Groebner basis of <tt>I</tt>. | ||

+ | <par/> | ||

<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||

<par/> | <par/> | ||

− | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ( | + | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>G</em>: a LIST of non-zero homogeneous polynomials generating a two-sided ideal in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1, | + | <item>@param <em>G</em>: a LIST of non-zero homogeneous polynomials generating a two-sided ideal in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].</item> |

<item>@param <em>DB</em>: a positive INT, which gives a degree bound of S-polynomials (or obstructions) during the Buchberger enumerating procedure. The procedure will discard S-polynomials (or obstructions) whose degrees are larger than DB.</item> | <item>@param <em>DB</em>: a positive INT, which gives a degree bound of S-polynomials (or obstructions) during the Buchberger enumerating procedure. The procedure will discard S-polynomials (or obstructions) whose degrees are larger than DB.</item> | ||

<item>@return: a LIST of polynomials, which is a truncated Groebner basis at degree DB with respect to the current word ordering if the enumerating procedure doesn't terminate due to reaching the loop bound LB, and is a partial Groebner basis otherwise.</item> | <item>@return: a LIST of polynomials, which is a truncated Groebner basis at degree DB with respect to the current word ordering if the enumerating procedure doesn't terminate due to reaching the loop bound LB, and is a partial Groebner basis otherwise.</item> | ||

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</itemize> | </itemize> | ||

<example> | <example> | ||

− | NCo.SetX( | + | NCo.SetX("xyz"); |

− | F1:=[[1, | + | F1:=[[1,"yxy"],[-1,"zyz"]]; |

− | F2:=[[1, | + | F2:=[[1,"xyx"],[-1,"zxy"]]; |

− | F3:=[[1, | + | F3:=[[1,"zxz"],[-1,"yzx"]]; |

− | F4:=[[1, | + | F4:=[[1,"xxx"],[1,"yyy"],[1,"zzz"],[1,"xyz"]]; |

G:=[F1,F2,F3,F4]; | G:=[F1,F2,F3,F4]; | ||

NCo.TruncatedGB(G,6); | NCo.TruncatedGB(G,6); | ||

− | [[[1, | + | [[[1, "yzx"], [-1, "zxz"]], [[1, "yxy"], [-1, "zyz"]], [[1, "xyx"], [-1, "zxy"]], [[1, "xxx"], [1, "xyz"], [1, "yyy"], [1, "zzz"]], |

− | [[1, | + | [[1, "zxzy"], [-1, "zzxz"]], [[1, "xzyz"], [-1, "zxyy"]], [[1, "xxyz"], [1, "xyyy"], [-1, "xzxz"], [1, "xzzz"], [-1, "yyyx"], |

− | [-1, | + | [-1, "zzzx"]], [[1, "zzxyy"], [-1, "zzxzz"]], [[1, "yzzxz"], [-1, "zxzzy"]], [[1, "yzzxy"], [-1, "zzxzx"]], [[1, "yzyyy"], |

− | [1, | + | [1, "yzzzz"], [1, "zxzxx"], [1, "zzxzz"]], [[1, "yxzxz"], [-1, "zyzzx"]], [[1, "xzzxz"], [-1, "zxyyx"]], [[1, "xyyyy"], [1, "xyzzz"], |

− | [1, | + | [1, "zxyyz"], [1, "zzzxy"]], [[1, "xxzxz"], [1, "xyyyx"], [-1, "xzxzx"], [1, "xzzzx"], [-1, "yyyxx"], [-1, "zzzxx"]], [[1, "xxzxy"], |

− | [1, | + | [1, "xyzyx"], [1, "yyyyx"], [1, "zzzyx"]], [[1, "xxyyy"], [1, "xxzzz"], [-1, "xyzyz"], [-1, "xzxzx"], [-1, "yyyxx"], [-1, "yyyyz"], |

− | [-1, | + | [-1, "zzzxx"], [-1, "zzzyz"]], [[1, "zxzzyz"], [-1, "zzxzxy"]], [[1, "yzzzxz"], [-1, "zxzzyy"]], [[1, "yzzzxy"], [-1, "zzxzxx"]], |

− | [[1, | + | [[1, "xzzzxz"], [-1, "zxyzyz"]], [[1, "xyyzyz"], [1, "xzxyyx"], [-1, "xzxzxy"], [1, "xzzzxy"], [-1, "yyyxxy"], [-1, "zzzxxy"]], |

− | [[1, | + | [[1, "xxzzzy"], [1, "xyyyzz"], [-1, "xyzyzy"], [-1, "xzxyyz"], [-1, "xzxzxy"], [-1, "xzxzzz"], [-1, "xzzzxy"], [1, "xzzzzz"], |

− | [-1, | + | [-1, "yyyxxy"], [-1, "yyyxzz"], [-1, "yyyyzy"], [-1, "zzzxxy"], [-1, "zzzxzz"], [-1, "zzzyzy"]], [[1, "xxzzxy"], [1, "xyzyxx"], |

− | [1, | + | [1, "yyyyxx"], [1, "zzzyxx"]]] |

------------------------------- | ------------------------------- | ||

</example> | </example> | ||

</description> | </description> | ||

<seealso> | <seealso> | ||

− | <see>NCo.GB</see> | + | <see>ApCoCoA-1:NCo.GB|NCo.GB</see> |

− | <see>NCo.IsGB</see> | + | <see>ApCoCoA-1:NCo.IsGB|NCo.IsGB</see> |

− | <see>NCo.ReducedGB</see> | + | <see>ApCoCoA-1:NCo.LW|NCo.LW</see> |

− | <see>NCo.SetFp</see> | + | <see>ApCoCoA-1:NCo.ReducedGB|NCo.ReducedGB</see> |

− | <see>NCo.SetOrdering</see> | + | <see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see> |

− | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> |

− | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> |

+ | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> | ||

</seealso> | </seealso> | ||

<types> | <types> | ||

Line 66: | Line 62: | ||

<key>NCo.TruncatedGB</key> | <key>NCo.TruncatedGB</key> | ||

<key>TruncatedGB</key> | <key>TruncatedGB</key> | ||

− | <wiki-category>Package_gbmr</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_gbmr</wiki-category> |

</command> | </command> |

## Latest revision as of 13:45, 29 October 2020

This article is about a function from ApCoCoA-1. |

## NCo.TruncatedGB

Compute a truncated Groebner basis of a finitely generated homogeneous two-sided ideal in a free monoid ring.

### Syntax

NCo.TruncatedGB(G:LIST, DB:INT):LIST

### Description

Given a word ordering and a homogeneous two-sided ideal `I`, a set of non-zero polynomials `G` is called a Groebner basis of `I` if the leading word set `LW{G}` generates the leading word ideal `LW(I)`. Note that it may not exist finite Groebner basis of the ideal `I`. Moreover, let `D` be a positive integer. Then the set `{g in G | Deg(g)<=D}` is a Groebner basis of the ideal `<f in I | Deg(f)<=D>` and is called a *D-truncated* Groebner basis of `I`.

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

Please set ring environment *coefficient field* ` K`, *alphabet* (or set of indeterminates) `X` and *ordering* via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is `Q`, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

@param

*G*: a LIST of non-zero homogeneous polynomials generating a two-sided ideal in`K<X>`. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in`<X>`and C is the coefficient of W. For example, the polynomial`f=xy-y+1`is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].@param

*DB*: a positive INT, which gives a degree bound of S-polynomials (or obstructions) during the Buchberger enumerating procedure. The procedure will discard S-polynomials (or obstructions) whose degrees are larger than DB.@return: a LIST of polynomials, which is a truncated Groebner basis at degree DB with respect to the current word ordering if the enumerating procedure doesn't terminate due to reaching the loop bound LB, and is a partial Groebner basis otherwise.

#### Example

NCo.SetX("xyz"); F1:=[[1,"yxy"],[-1,"zyz"]]; F2:=[[1,"xyx"],[-1,"zxy"]]; F3:=[[1,"zxz"],[-1,"yzx"]]; F4:=[[1,"xxx"],[1,"yyy"],[1,"zzz"],[1,"xyz"]]; G:=[F1,F2,F3,F4]; NCo.TruncatedGB(G,6); [[[1, "yzx"], [-1, "zxz"]], [[1, "yxy"], [-1, "zyz"]], [[1, "xyx"], [-1, "zxy"]], [[1, "xxx"], [1, "xyz"], [1, "yyy"], [1, "zzz"]], [[1, "zxzy"], [-1, "zzxz"]], [[1, "xzyz"], [-1, "zxyy"]], [[1, "xxyz"], [1, "xyyy"], [-1, "xzxz"], [1, "xzzz"], [-1, "yyyx"], [-1, "zzzx"]], [[1, "zzxyy"], [-1, "zzxzz"]], [[1, "yzzxz"], [-1, "zxzzy"]], [[1, "yzzxy"], [-1, "zzxzx"]], [[1, "yzyyy"], [1, "yzzzz"], [1, "zxzxx"], [1, "zzxzz"]], [[1, "yxzxz"], [-1, "zyzzx"]], [[1, "xzzxz"], [-1, "zxyyx"]], [[1, "xyyyy"], [1, "xyzzz"], [1, "zxyyz"], [1, "zzzxy"]], [[1, "xxzxz"], [1, "xyyyx"], [-1, "xzxzx"], [1, "xzzzx"], [-1, "yyyxx"], [-1, "zzzxx"]], [[1, "xxzxy"], [1, "xyzyx"], [1, "yyyyx"], [1, "zzzyx"]], [[1, "xxyyy"], [1, "xxzzz"], [-1, "xyzyz"], [-1, "xzxzx"], [-1, "yyyxx"], [-1, "yyyyz"], [-1, "zzzxx"], [-1, "zzzyz"]], [[1, "zxzzyz"], [-1, "zzxzxy"]], [[1, "yzzzxz"], [-1, "zxzzyy"]], [[1, "yzzzxy"], [-1, "zzxzxx"]], [[1, "xzzzxz"], [-1, "zxyzyz"]], [[1, "xyyzyz"], [1, "xzxyyx"], [-1, "xzxzxy"], [1, "xzzzxy"], [-1, "yyyxxy"], [-1, "zzzxxy"]], [[1, "xxzzzy"], [1, "xyyyzz"], [-1, "xyzyzy"], [-1, "xzxyyz"], [-1, "xzxzxy"], [-1, "xzxzzz"], [-1, "xzzzxy"], [1, "xzzzzz"], [-1, "yyyxxy"], [-1, "yyyxzz"], [-1, "yyyyzy"], [-1, "zzzxxy"], [-1, "zzzxzz"], [-1, "zzzyzy"]], [[1, "xxzzxy"], [1, "xyzyxx"], [1, "yyyyxx"], [1, "zzzyxx"]]] -------------------------------

### See also