Difference between revisions of "ApCoCoA-1:NC.Intersection"
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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 ring environment coefficient field <tt>K</tt> | + | Please set ring environment coefficient field <tt>K</tt> and alphabet (or indeterminates) <tt>X</tt> through the functions <ref>NC.SetFp</ref>(Prime) and <ref>NC.SetX</ref>(X), 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>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> | <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> | ||
Revision as of 22:01, 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 and alphabet (or indeterminates) X through the functions NC.SetFp(Prime) and NC.SetX(X), 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
