# Difference between revisions of "ApCoCoA-1:NC.KernelOfHomomorphism"

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<title>NC.KernelOfHomomorphism</title> | <title>NC.KernelOfHomomorphism</title> | ||

<short_description> | <short_description> | ||

− | ( | + | Enumerate a (partial) Groebner basis of the kernel of a <tt>K</tt>-algebra homomorphism. |

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

<syntax> | <syntax> | ||

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<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 | + | Please set the <em>coefficient field</em> <tt>K</tt> via the functions <ref>NC.SetFp</ref> (or <ref>NC.UnsetFp</ref>) before calling the function. Default coefficient field is <tt>Q</tt>. For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>X1:</em> a finite set of | + | <item>@param <em>X1:</em> a finite alphabet or finite set of indeterminates. It is of STRING type. Note that every letter in X1 MUST appear only once.The order of letters in X1 is very important, since <tt>X1</tt> together with <tt>Images</tt> determinates a K-algebra homomorphism.</item> |

− | <item>@param <em>X2:</em> another finite set of | + | <item>@param <em>X2:</em> another finite alphabet or finite set of indeterminates. It is of STRING type. Note that X1 and X2 are disjoint.</item> |

− | <item>@param <em>Images:</em> a LIST of polynomials in <tt>K<X2></tt>. Each polynomial is represented as a LIST of | + | <item>@param <em>Images:</em> a LIST of polynomials in <tt>K<X2></tt>. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <tt><X2></tt> and C is the coefficient of W. For example, the polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. The zero polynomial <tt>0</tt> is represented as an empty LIST []. Note that the number of polynomials in <tt>Images</tt> should be equal to the number of letters in <tt>X1</tt>. Also note that the order of polynomials in Images is very important, since <tt>Images</tt> together with <tt>X1</tt> defines a K-algebra homomorphism. For example, X1:=<quotes>abc</quotes>; X2:=<quotes>xy</quotes>; F1 := [[1,<quotes>x</quotes>], [1,<quotes>y</quotes>]]; F2 := [[1,<quotes>xx</quotes>],[1,<quotes>xy</quotes>]]; F3 := [[1,<quotes>yy</quotes>],[1,<quotes>yx</quotes>]]; Images := [F1, F2, F3]; then the K-algebra homomorphism is defined by mapping a to F1, b to F2 and c to F3.</item> |

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

− | Since | + | Since this function is based on Groebner basis computations, we refer users to <ref>NC.GB</ref> or <ref>NC.ReducedGB</ref> for information about the following three optional parameters: |

<itemize> | <itemize> | ||

<item>@param <em>DegreeBound</em></item> | <item>@param <em>DegreeBound</em></item> | ||

<item>@param <em>LoopBound</em></item> | <item>@param <em>LoopBound</em></item> | ||

<item>@param <em>Flag</em></item> | <item>@param <em>Flag</em></item> | ||

− | <item>@return: a LIST of polynomials, which is a Groebner basis of the kernel of | + | <item>@return: a LIST of polynomials, which is a Groebner basis of the kernel of the <tt>K</tt>-algebra homomorphism if a finite Groebner basis exists, and is a partial Groebner basis otherwise.</item> |

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

<example> | <example> | ||

Line 42: | Line 42: | ||

<see>NC.GB</see> | <see>NC.GB</see> | ||

<see>NC.HF</see> | <see>NC.HF</see> | ||

+ | <see>NC.Interreduction</see> | ||

<see>NC.Intersection</see> | <see>NC.Intersection</see> | ||

<see>NC.IsGB</see> | <see>NC.IsGB</see> |

## Revision as of 14:48, 7 June 2012

## NC.KernelOfHomomorphism

Enumerate a (partial) Groebner basis of the kernel of a `K`-algebra homomorphism.

### Syntax

NC.KernelOfHomomorphism(X1:STRING, X2:STRING, Images:LIST):LIST NC.KernelOfHomomorphism(X1:STRING, X2:STRING, Images:LIST, DegreeBound:INT, LoopBound:INT, Flag:INT):LIST

### Description

*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 the *coefficient field* `K` via the functions NC.SetFp (or NC.UnsetFp) before calling the function. Default coefficient field is `Q`. For more information, please check the relevant functions.

@param

*X1:*a finite alphabet or finite set of indeterminates. It is of STRING type. Note that every letter in X1 MUST appear only once.The order of letters in X1 is very important, since`X1`together with`Images`determinates a K-algebra homomorphism.@param

*X2:*another finite alphabet or finite set of indeterminates. It is of STRING type. Note that X1 and X2 are disjoint.@param

*Images:*a LIST of polynomials in`K<X2>`. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in`<X2>`and C is the coefficient of W. For example, the polynomial`F=xy-y+1`is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial`0`is represented as an empty LIST []. Note that the number of polynomials in`Images`should be equal to the number of letters in`X1`. Also note that the order of polynomials in Images is very important, since`Images`together with`X1`defines a K-algebra homomorphism. For example, X1:="abc"; X2:="xy"; F1 := [[1,"x"], [1,"y"]]; F2 := [[1,"xx"],[1,"xy"]]; F3 := [[1,"yy"],[1,"yx"]]; Images := [F1, F2, F3]; then the K-algebra homomorphism is defined by mapping a to F1, b to F2 and c to F3.

Since this function is based on Groebner basis computations, we refer users to NC.GB or NC.ReducedGB for information about the following three optional parameters:

@param

*DegreeBound*@param

*LoopBound*@param

*Flag*@return: a LIST of polynomials, which is a Groebner basis of the kernel of the

`K`-algebra homomorphism if a finite Groebner basis exists, and is a partial Groebner basis otherwise.

#### Example

X1 := <quotes>abc</quotes>; X2 := <quotes>xy</quotes>; F1 := [[1,<quotes>x</quotes>], [1,<quotes>y</quotes>]]; F2 := [[1,<quotes>xx</quotes>],[1,<quotes>xy</quotes>]]; F3 := [[1,<quotes>yy</quotes>],[1,<quotes>yx</quotes>]]; Images :=[F1, F2, F3]; -- a |-> F1; b |-> F2; c |-> F3 NC.KernelOfHomomorphism(X1, X2, Images); [[[1, <quotes>ab</quotes>], [-1, <quotes>ba</quotes>], [1, <quotes>ac</quotes>], [-1, <quotes>ca</quotes>]], [[1, <quotes>aa</quotes>], [-1, <quotes>b</quotes>], [-1, <quotes>c</quotes>]]] -------------------------------

### See also