Difference between revisions of "ApCoCoA-1:NC.IsHomog"

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.IsHomog</title>
 
<title>NC.IsHomog</title>
 
<short_description>
 
<short_description>
−
Check whether a polynomial of a list of polynomials is homogeneous over a free monoid ring.
+
Check whether a polynomial or a LIST of polynomials is homogeneous in a non-commutative polynomial ring.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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</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.
+
Please set non-commutative polynomial ring (via the command <ref>ApCoCoA-1:Use|Use</ref>) before calling this function. For more information, please check the relevant commands and functions.
−
<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>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.SetOrdering</ref>, respectively, before calling the function. The default coefficient field is <tt>Q</tt>. The default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
 
 
<itemize>
 
<itemize>
−
<item>@param <em>F</em>: a polynomial or a LIST of polynomials in <tt>K&lt;X&gt;</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>&lt;X&gt;</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 the empty LIST [].</item>
+
<item>@param <em>F</em>: a non-commutative polynomial or a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
−
<item>@return: a BOOL value which is True if F is homogeneous and False otherwise. Note that if F is a set of homogeneous polynomials, then F generates a homogeneous ideal. It is false contrarily.</item>
+
<item>@return: a BOOL, which is True if F is homogeneous and False otherwise.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
−
NC.SetX(<quotes>xy</quotes>);  
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USE QQ[x[1..2],y[1..2]];
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F1 := [[1,<quotes>x</quotes>], [1,<quotes>y</quotes>]];
+
F1:= [[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3]]; -- 2x[1]y[1]x[2]^2-9y[2]x[1]^2x[2]^3
−
F2 := [[1,<quotes>xx</quotes>],[1,<quotes>xy</quotes>],[1,<quotes>x</quotes>]];  
+
F2:= [[2x[1],y[1],x[2]^2], [y[2],x[2]^3]]; -- 2x[1]y[1]x[2]^2+y[2]x[2]^3
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F := [F1,F2];  
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F3:= [[2x[1],y[1],x[2]]]; -- 2x[1]y[1]x[2]
−
NC.IsHomog(F);
+
NC.IsHomog(F1);
 +
NC.IsHomog(F2);
 +
NC.IsHomog(F3);
 +
NC.IsHomog([F1,F2,F3]);
 +
NC.IsHomog([F2,F3]);
 +
 
 
False
 
False
−
 
 
-------------------------------
 
-------------------------------
−
NC.IsHomog(F1);
 
 
True
 
True
−
 
 
-------------------------------
 
-------------------------------
−
NC.IsHomog(F2);
+
True
 +
-------------------------------
 
False
 
False
 +
-------------------------------
 +
True
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
−
<see>NC.Add</see>
+
<see>ApCoCoA-1:Use|Use</see>
−
<see>NC.Deg</see>
 
−
<see>NC.FindPolynomials</see>
 
−
<see>NC.GB</see>
 
−
<see>NC.HF</see>
 
−
<see>NC.Interreduction</see>
 
−
<see>NC.Intersection</see>
 
−
<see>NC.IsGB</see>
 
−
<see>NC.IsHomog</see>
 
−
<see>NC.KernelOfHomomorphism</see>
 
−
<see>NC.LC</see>
 
−
<see>NC.LT</see>
 
−
<see>NC.LTIdeal</see>
 
−
<see>NC.MB</see>
 
−
<see>NC.MinimalPolynomial</see>
 
−
<see>NC.Multiply</see>
 
−
<see>NC.NR</see>
 
−
<see>NC.ReducedGB</see>
 
−
<see>NC.SetFp</see>
 
−
<see>NC.SetOrdering</see>
 
−
<see>NC.SetRelations</see>
 
−
<see>NC.SetRules</see>
 
−
<see>NC.SetX</see>
 
−
<see>NC.Subtract</see>
 
−
<see>NC.UnsetFp</see>
 
−
<see>NC.UnsetOrdering</see>
 
−
<see>NC.UnsetRelations</see>
 
−
<see>NC.UnsetRules</see>
 
−
<see>NC.UnsetX</see>
 
−
<see>Introduction to CoCoAServer</see>
 
 
</seealso>
 
</seealso>
 
<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
−
<type>groebner</type>
+
<type>polynomial</type>
−
<type>ideal</type>
 
 
<type>non_commutative</type>
 
<type>non_commutative</type>
 
</types>
 
</types>
−
<key>gbmr.IsHomog</key>
+
<key>ncpoly.IsHomog</key>
 
<key>NC.IsHomog</key>
 
<key>NC.IsHomog</key>
 
<key>IsHomog</key>
 
<key>IsHomog</key>
−
<wiki-category>Package_gbmr</wiki-category>
+
<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 10:14, 7 October 2020

This article is about a function from ApCoCoA-1.

NC.IsHomog

Check whether a polynomial or a LIST of polynomials is homogeneous in a non-commutative polynomial ring.

Syntax

NC.IsHomog(F:LIST):BOOL

Description

Please set non-commutative polynomial ring (via the command Use) before calling this function. For more information, please check the relevant commands and functions.

  • @param F: a non-commutative polynomial or a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial 0 is represented as the empty LIST [].

  • @return: a BOOL, which is True if F is homogeneous and False otherwise.

Example

USE QQ[x[1..2],y[1..2]];
F1:= [[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3]]; -- 2x[1]y[1]x[2]^2-9y[2]x[1]^2x[2]^3
F2:= [[2x[1],y[1],x[2]^2], [y[2],x[2]^3]]; -- 2x[1]y[1]x[2]^2+y[2]x[2]^3
F3:= [[2x[1],y[1],x[2]]]; -- 2x[1]y[1]x[2]
NC.IsHomog(F1);
NC.IsHomog(F2);
NC.IsHomog(F3);
NC.IsHomog([F1,F2,F3]);
NC.IsHomog([F2,F3]);

False
-------------------------------
True
-------------------------------
True
-------------------------------
False
-------------------------------
True
-------------------------------

See also

Use




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