Difference between revisions of "Package sagbi/SB.TruncSAGBI"
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<command> | <command> | ||
<title>SB.TruncSAGBI</title> | <title>SB.TruncSAGBI</title> | ||
| − | <short_description>Computes a truncated SAGBI-basis of a subalgebra.</short_description> | + | <short_description>Computes a truncated SAGBI-basis of a standard-graded subalgebra.</short_description> |
<syntax> | <syntax> | ||
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This function computes a <tt>d</tt>-truncated SAGBI-basis of the subalgebra <tt>S</tt> generated by the polynomials in the list <tt>G</tt>, i.e. a set of polynomials <tt>F</tt> such that any term appearing as the leading term of a polynomial in <tt>S</tt> of degree smaller than or equal to <tt>d</tt> is a product of terms appearing as leading term of a polynomial in <tt>F</tt>. | This function computes a <tt>d</tt>-truncated SAGBI-basis of the subalgebra <tt>S</tt> generated by the polynomials in the list <tt>G</tt>, i.e. a set of polynomials <tt>F</tt> such that any term appearing as the leading term of a polynomial in <tt>S</tt> of degree smaller than or equal to <tt>d</tt> is a product of terms appearing as leading term of a polynomial in <tt>F</tt>. | ||
<itemize> | <itemize> | ||
| − | <item>@param <em>G</em> A list of polynomials which generates a subalgebra.</item> | + | <item>@param <em>G</em> A list of homogeneous polynomials which generates a subalgebra.</item> |
<item>@param <em>d</em> An integer specifying the truncation degree.</item> | <item>@param <em>d</em> An integer specifying the truncation degree.</item> | ||
<item>@return A tuple <tt>[L,b]</tt> where <tt>L</tt> is a list of polynomials which form a <tt>d</tt>-truncated finite SAGBI-basis of the subalgebra generated by <tt>G</tt>. <tt>b</tt> is a boolean value which is <tt>true</tt> if <tt>L</tt> is already a complete SAGBI basis.</item> | <item>@return A tuple <tt>[L,b]</tt> where <tt>L</tt> is a list of polynomials which form a <tt>d</tt>-truncated finite SAGBI-basis of the subalgebra generated by <tt>G</tt>. <tt>b</tt> is a boolean value which is <tt>true</tt> if <tt>L</tt> is already a complete SAGBI basis.</item> | ||
Revision as of 10:44, 27 September 2020
| This article is about a function in ApCoCoA-2.0. |
SB.TruncSAGBI
Computes a truncated SAGBI-basis of a standard-graded subalgebra.
Syntax
SB.SAGBI(G:LIST of POLY,d:RAT):LIST of POLY
Description
This function computes a d-truncated SAGBI-basis of the subalgebra S generated by the polynomials in the list G, i.e. a set of polynomials F such that any term appearing as the leading term of a polynomial in S of degree smaller than or equal to d is a product of terms appearing as leading term of a polynomial in F.
@param G A list of homogeneous polynomials which generates a subalgebra.
@param d An integer specifying the truncation degree.
@return A tuple [L,b] where L is a list of polynomials which form a d-truncated finite SAGBI-basis of the subalgebra generated by G. b is a boolean value which is true if L is already a complete SAGBI basis.
Example
Use QQ[x[1..3]];
S := SB.TruncSAGBI([x[1]^2-x[3]^2,x[1]*x[2]+x[3]^2,x[2]^2-2*x[3]^2],2);
indent(S,2);
-----------------------------------------------------------------------------
[
[
x[2]^2 -2*x[3]^2,
x[1]*x[2] +x[3]^2,
x[1]^2 -x[3]^2,
x[1]^2*x[3]^2 +x[1]*x[2]*x[3]^2 +(1/2)*x[2]^2*x[3]^2 +(-1/2)*x[3]^4
],
true
]
Example
Use QQ[x,y];
S := SB.TruncSAGBI([x+y,x*y,x*y^2],10); -- K[x+y,xy,xy^2] doesn't have a finite SAGBI basis
indent(S,2);
-----------------------------------------------------------------------------
[
[
x +y,
x*y,
x*y^2,
x*y^3,
x*y^4
],
false
]
