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7.10.4.5 lpDivision
Procedure from library freegb.lib (see freegb_lib).
- Usage:
- lpDivision(p,G); poly p, ideal G
- Purpose:
- compute a two-sided division with remainder of p wrt G; two-sided noncommutative analogue of the procedure division
- Assume:
- G = {g1,...,gN} is a Groebner basis, the original ring of the Letterplace ring has the name 'r' and no variable is called 'tag_i' for i in 1...N
- Return:
- list L
- Note:
- - L[1] is NF(p,I)
- L[2] is the list of expressions [i,l_(ij),r_(ij)] with \sum_(i,j) l_(ij) g_i r_(ij) = p - NF(p,I)
- procedure lpGBPres2Poly, applied to L, reconstructs p
Example:
| | LIB "freegb.lib";
ring r = 0,(x,y),dp;
ring R = freeAlgebra(r, 4);
ideal I = x*x + y*y - 1; // 2D sphere
ideal J = twostd(I); // compute a two-sided Groebner basis
J; // it is finite and nice
==> J[1]=x*x+y*y-1
==> J[2]=y*y*x-x*y*y
poly h = x*x*y-y*x*x+x*y;
list L = lpDivision(h,J); L; // what means that the NF of h wrt J is x*y
==> [1]:
==> x*y
==> [2]:
==> [1]:
==> [1]:
==> 1
==> [2]:
==> 1
==> [3]:
==> y
==> [2]:
==> [1]:
==> 1
==> [2]:
==> -y
==> [3]:
==> 1
h - lpNF(h,J); // and this poly has the following two-sided Groebner presentation:
==> -y*x*x+x*x*y
-y*J[1] + J[1]*y;
==> -y*x*x+x*x*y
lpGBPres2Poly(L,J); // reconstructs the above automatically
==> -y*x*x+x*x*y+x*y
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