Synopsis

• Usage:
cssLeadTerm(I, w)
• Inputs:
• I, an ideal, (regular) holonomic ideal in the Weyl algebra
• w, a list, (generic) weights for $I$, of length half the number of variables in the Weyl algebra
• Outputs:
• a list, containing monomials times logarithms in the variables

Description

This routine returns the lead terms of the canonical series solutions of $I$ with respect to the weight vector $w$. See [SST, Algorithm 2.3.14 and Lemma 2.5.10].

Here is [SST, Example 2.3.16]:

 i1 : needsPackage "FourTiTwo" o1 = FourTiTwo o1 : Package i2 : A = matrix{{1,1,1,1,1,1},{-2,0,0,0,0,1},{0,1,0,1,0,0},{1,1,2,0,0,1}} o2 = | 1 1 1 1 1 1 | | -2 0 0 0 0 1 | | 0 1 0 1 0 0 | | 1 1 2 0 0 1 | 4 6 o2 : Matrix ZZ <-- ZZ i3 : beta = {2,1,0,2} o3 = {2, 1, 0, 2} o3 : List i4 : Hbeta = gkz(A,beta) o4 = ideal (x D + x D + x D + x D + x D + x D - 2, - 2x D + x D - 1, 1 1 2 2 3 3 4 4 5 5 6 6 1 1 6 6 ------------------------------------------------------------------------ 2 2 2 x D + x D , x D + x D + 2x D + x D - 2, - D D + D D , - D D D + 2 2 4 4 1 1 2 2 3 3 6 6 3 4 2 5 2 3 5 ------------------------------------------------------------------------ 2 2 2 3 3 2 4 D D D , - D D D + D D D , - D D + D D ) 1 4 6 3 4 5 1 2 6 3 5 1 6 o4 : Ideal of QQ[x ..x , D ..D ] 1 6 1 6 i5 : w = {9,1,99999, 9999999, 3, 999} o5 = {9, 1, 99999, 9999999, 3, 999} o5 : List i6 : netList cssLeadTerm(Hbeta, w) -- 6.44e-06 seconds elapsed -- 6.69e-06 seconds elapsed -- 6.7e-06 seconds elapsed -- 9.151e-06 seconds elapsed -- 6.591e-06 seconds elapsed Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal. Converting to Naive algorithm. +----------------------------------------------------+ | 1 5 5 5 | | - - - - - - | | 2 2 2 2 | o6 = |x x x x | | 1 2 4 5 | +----------------------------------------------------+ | -1 | |x x x x | | 2 3 4 6 | +----------------------------------------------------+ | -1 1 3 3 | |x x x x (-logX - -logX - -logX + logX ) | | 2 3 4 6 2 0 4 2 4 4 5 | +----------------------------------------------------+ | -1 | |x x x x | | 2 4 5 6 | +----------------------------------------------------+ | -1 1 3 3 3 | |x x x x (-logX - -logX + -logX - -logX + logX )| | 2 4 5 6 2 0 2 1 2 3 2 4 5 | +----------------------------------------------------+ | -1 2 2 -1 | |x x x x | | 1 3 5 6 | +----------------------------------------------------+ | -1 2 2 -1 1 3 3 | |x x x x (-logX - -logX - -logX + logX ) | | 1 3 5 6 2 0 4 2 4 4 5 | +----------------------------------------------------+ | 1 5 | | - - | | 3 3 | |x x | | 1 6 | +----------------------------------------------------+ | 1 5 | | - - | | 3 3 1 3 3 | |x x (-logX - -logX - -logX + logX ) | | 1 6 2 0 4 2 4 4 5 | +----------------------------------------------------+ | 1 1 | | - - - | | 3 3 | |x x x x | | 1 3 5 6 | +----------------------------------------------------+ | 1 1 | | - - - | | 3 3 1 3 3 | |x x x x (-logX - -logX - -logX + logX ) | | 1 3 5 6 2 0 4 2 4 4 5 | +----------------------------------------------------+