# ApCoCoA-1:BB.LiftNDViaServer

## BB.LiftNDViaServer

Compute the border basis scheme ideal generators obtained from lifting of ND neighbors.

### Syntax

BB.LiftNDViaServer(OO:LIST,Border:LIST,HomogeneousLift:BOOL):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.

If HomogeneousLift is set to False, the generators of the border basis scheme ideal <formula>I(\mathbb{B}_\mathcal{O})</formula> that result from the lifting of next-door neighbors will computed by using the ApCoCoAServer. The input is a list of terms OO representing an order ideal and a list of terms Border representing the border of the order ideal. If HomogeneousLift is set to True, generators of <formula>I(\mathbb{B}^{\textrm{hom}}_\mathcal{O})</formula> will be computed instead. The output is a list of polynomials in the ring <formula>BBS=K[c_{ij}]</formula>.

• @param OO A list of terms representing an order ideal.

• @param Border A list of terms representing the border of OO

• @param Homogeneous Set to TRUE if you want to compute the generators of the homogeneous border basis scheme.

• @return A list of generators of the border basis scheme ideal I(B_O) that results from the lifting of next-door neighbors. The polynomials will belong to the ring BBS=K[c_{ij}].

#### Example

Use Q[x,y], DegRevLex;
BB.LiftNDViaServer([Poly(1), x, y, xy], [y^2, x^2, xy^2, x^2y], False);

-------------------------------
[BBS :: c[2,1]c[4,2] + c[4,1]c[4,4] + c[3,1] - c[4,3],
BBS :: c[2,1]c[2,2] + c[2,4]c[4,1] + c[1,1] - c[2,3],
BBS :: c[2,1]c[3,2] + c[3,4]c[4,1] - c[3,3],
BBS :: c[1,2]c[2,1] + c[1,4]c[4,1] - c[1,3],
BBS :: c[3,2]c[4,1] + c[4,2]c[4,3] + c[2,2] - c[4,4],
BBS :: c[2,1]c[3,2] + c[2,3]c[4,2] - c[2,4],
BBS :: c[3,1]c[3,2] + c[3,3]c[4,2] + c[1,2] - c[3,4],
BBS :: c[1,1]c[3,2] + c[1,3]c[4,2] - c[1,4]]
-------------------------------

Use Q[x,y,z], DegRevLex;
BB.LiftNDViaServer([Poly(1), x, y, xy], [z, yz, xz, y^2, x^2, xyz, xy^2, x^2y], True);

-------------------------------
[BBS :: c[3,1]c[4,4] + c[2,1] - c[4,2],
BBS :: c[2,1]c[4,5] + c[3,1] - c[4,3]]
-------------------------------