Difference between revisions of "ApCoCoA-1:NC.Intersection"
| Line 2: | Line 2: | ||
<title>NC.Intersection</title> | <title>NC.Intersection</title> | ||
<short_description> | <short_description> | ||
| − | + | (Partial) Groebner basis of the intersection of two finitely generated two-sided ideals over a free associative <tt>K</tt>-algebra. | |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
| − | NC.Intersection( | + | NC.Intersection(G1:LIST, G2:LIST):LIST |
| − | NC.Intersection( | + | NC.Intersection(G1:LIST, G2:LIST, DegreeBound:INT, LoopBound:INT, Flag:INT):LIST |
</syntax> | </syntax> | ||
<description> | <description> | ||
<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 coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions. | |
<itemize> | <itemize> | ||
| − | <item>@param <em> | + | <item>@param <em>G1, G2:</em> two LISTs of non-zero polynomials and each generates a two-sided ideal in <tt>K<X></tt>. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item> |
| − | |||
| − | |||
</itemize> | </itemize> | ||
| − | Since the algorithm used in this function is based on Groebner basis computation, we refer users to <ref>NC. | + | Since the algorithm used in this function is based on Groebner basis computation, we refer users to <ref>NC.GB</ref> or <ref>NC.ReducedGB</ref> for information about the following 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 intersection of (G1) and (G2) if a finite Groebner basis exists, and is a partial Groebner basis set otherwise.</item> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
Revision as of 21:55, 11 December 2010
NC.Intersection
(Partial) Groebner basis of the intersection of two finitely generated two-sided ideals over a free associative K-algebra.
Syntax
NC.Intersection(G1:LIST, G2:LIST):LIST NC.Intersection(G1:LIST, G2: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 ring environment coefficient field K, alphabet (or indeterminates) X and ordering through the functions NC.SetFp(Prime), NC.SetX(X) and NC.SetOrdering(Ordering), respectively, before calling the function. Default coefficient field is Q. Default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.
@param G1, G2: two LISTs of non-zero polynomials and each generates a two-sided ideal in K<X>. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, xy^2x is represented as "xyyx", unit is represented as an empty string "". Then, polynomial F=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. 0 polynomial is represented as an empty LIST [].
Since the algorithm used in this function is based on Groebner basis computation, we refer users to NC.GB or NC.ReducedGB for information about the following optional parameters:
@param DegreeBound
@param LoopBound
@param Flag
@return: a LIST of polynomials, which is a Groebner basis of the intersection of (G1) and (G2) if a finite Groebner basis exists, and is a partial Groebner basis set otherwise.
Example
NC.SetFp(); -- set default Fp=F2
NC.SetX(<quotes>xyz</quotes>);
F1 := [[1,<quotes>xy</quotes>], [1,<quotes>z</quotes>]];
F2 := [[1,<quotes>yz</quotes>], [1, <quotes>x</quotes>]];
F3 := [[1,<quotes>zx</quotes>], [1,<quotes>y</quotes>]];
Ideal_I := [F1, F2]; -- ideal generated by {xy+z, yz+x}
Ideal_J := [F2, F3]; -- ideal generated by {yz+x, zx+y}
NC.Intersection(Ideal_I, Ideal_J, 20, 25, 1);
[[[1, <quotes>zyzzz</quotes>], [1, <quotes>zzzyz</quotes>], [1, <quotes>yzz</quotes>], [1, <quotes>zzy</quotes>]], [[1, <quotes>yzyz</quotes>], [1, <quotes>zyzy</quotes>]], [[1, <quotes>zyzyyz</quotes>], [1, <quotes>yzyy</quotes>], [1, <quotes>zzyz</quotes>], [1, <quotes>zy</quotes>]],
[[1, <quotes>yzzyzy</quotes>], [1, <quotes>yzyy</quotes>], [1, <quotes>zzyz</quotes>], [1, <quotes>zy</quotes>]], [[1, <quotes>zzzzyzyy</quotes>], [1, <quotes>zzyyzy</quotes>], [1, <quotes>zzzyzz</quotes>], [1, <quotes>zzyz</quotes>]],
[[1, <quotes>zzyzyyyyz</quotes>], [1, <quotes>zyzyyyy</quotes>], [1, <quotes>yzzzyzy</quotes>], [1, <quotes>zzzyyyz</quotes>], [1, <quotes>yzyyz</quotes>], [1, <quotes>zzyyy</quotes>], [1, <quotes>zzyzz</quotes>], [1, <quotes>zyz</quotes>]],
[[1, <quotes>x</quotes>], [1, <quotes>yz</quotes>]]]
-------------------------------
Note the following information printed by the server shows it is a partial Groebner basis.
===== 25th Loop =====
Number of elements in (partial) Groebner basis G: 19 -- partial Groebner basis before being interreduced
Number of S-elements: 25/113 -- 25 S-elements have been check, and 113 unchecked S-elements
See also
