Kronecker Package: Description

The solution of Problem~#GSS#556> is incremental on the number of equations to be solved, thus iteratively solving each system #math94##tex2html_wrap_inline4532#, each resolution being encoded by means of a lifting fiber.

Thus, at each step the algorithm depends on the choice of a Noether position for #math95##tex2html_wrap_inline4536#, a lifting point and a primitive element, Zariski-openness granting that such choices can be done randomly.

Let us therefore assume we have

<#4537#><#4537#>
the #math96#δ = r - ρ-dimensional variety #math97##tex2html_wrap_inline4548#⊂#tex2html_wrap_inline4549#Z(f1,…, fr),
<#4538#><#4538#>
the projection #math98#φ : #tex2html_wrap_inline4553# #tex2html_wrap_inline4554# #tex2html_wrap_inline4557# defined by #math99#φ(a1,…, ar) = (a1,…, aδ),
<#4539#><#4539#>
the lifting system #math100#f1,…, fρ of #math101##tex2html_wrap_inline4563#,
a frame of coordinates which is in Noether position for #math102##tex2html_wrap_inline4567# and which, by simplicity, we assume to be #math103#{Z1,…, Zr},
<#4540#><#4540#>
a lifting point #math104##tex2html_wrap_inline4570# : = (p1,…, pδ) of #math105##tex2html_wrap_inline4574# w.r.t. the lifting system #math106#f1,…, fρ and the frame #math107#{Z1,…, Zr},
<#4541#><#4541#>
the primitive element #math108#U : = λδ+1Zδ+1 + ... + λrZr, λi∈k, λδ+1≠0, of #math109#K[Zδ+1,…, Zr]/#tex2html_wrap_inline4581#,
<#4542#><#4542#>
the minimal polynomial q(T) of U,
<#4543#><#4543#>
the parametrization #math110#(vδ+1(T), vδ+2(T),…, vr(T)) of both #math111##tex2html_wrap_inline4588# and #math112##tex2html_wrap_inline4592#.

Up to now we simply assume that

are sufficiently generic in order to satisfy all the conditions of genericity required by the algorithm; we will discuss deepler such conditions in Section~#43S8#595>.

<#4597#>Lifting Step<#4597#>
Thus we are assuming to have a geometric resolution

#math116#

#tex2html_wrap_indisplay4601##tex2html_wrap_indisplay4602#

for the primitive element

#math117#

U : = λδ+1Zδ+1 + ... + λrZr∈K[Zδ+1,…, Zr]/#tex2html_wrap_indisplay4606#

of the variety #math118##tex2html_wrap_inline4610# defined by

#math119#

#tex2html_wrap_indisplay4612# : = (p1,…, pδ, αδ+1,…, αr)∈#tex2html_wrap_indisplay4615##tex2html_wrap_indisplay4616#f1(#tex2html_wrap_indisplay4617#) = … = fρ(#tex2html_wrap_indisplay4618#) = 0≠g(#tex2html_wrap_indisplay4619#)

and the 0-dimensional radical ideal

#math120#

#tex2html_wrap_indisplay4623# : = #tex2html_wrap_indisplay4624#I(#tex2html_wrap_indisplay4627#) = #tex2html_wrap_indisplay4630# + (Z1 - p1,…, Zδ - pδ)

and we compute a geometric resolution

#math121#

#tex2html_wrap_indisplay4632##tex2html_wrap_indisplay4633#

for the primitive element

#math122#

U : = #tex2html_wrap_indisplay4635#λδ+iZδ+i∈k(Zδ)[Zδ+1,…, Zr]/#tex2html_wrap_indisplay4638#

of the variety #tex2html_wrap_inline4642# defined by

#math123#

#tex2html_wrap_indisplay4644# = (p1,…, pδ-1, αδ,…, αr)∈#tex2html_wrap_indisplay4647##tex2html_wrap_indisplay4648#f1(#tex2html_wrap_indisplay4649#) = ... = fρ(#tex2html_wrap_indisplay4650#) = 0≠g(#tex2html_wrap_indisplay4651#)

and the 1-dimensional radical ideal

#math124#

#tex2html_wrap_indisplay4655# : = #tex2html_wrap_indisplay4656#I(#tex2html_wrap_indisplay4659#) = #tex2html_wrap_indisplay4662# + (Z1 - p1,…, Zδ-1 - pδ-1).

<#4598#>Intersection Step<#4598#>
From this date we compute a geometric resolution

#math125#

#tex2html_wrap_indisplay4664##tex2html_wrap_indisplay4665#

for the primitive element

#math126#

U : = #tex2html_wrap_indisplay4667#λδ+jZδ+j∈32K[Zδ,…, Zr]/#tex2html_wrap_indisplay4669#

of the 0-dimensional radical ideal

#math127#

#tex2html_wrap_indisplay4672# : = #tex2html_wrap_indisplay4673#.

<#4599#>Cleaning Step<#4599#>
We now remove the points #math128##tex2html_wrap_inline4675#∈#tex2html_wrap_inline4676#Z(#tex2html_wrap_inline4678#) such that #math129#g(#tex2html_wrap_inline4680#) = 0 thus getting the required geometric resolution

#math130#

#tex2html_wrap_indisplay4682##tex2html_wrap_indisplay4683#

for the primitive element

#math131#

U : = λδZδ + ... + λrZr∈K[Zδ,…, Zr]/#tex2html_wrap_indisplay4687#

and the lifting point #math132##tex2html_wrap_inline4690# : = (p1,…, pδ-1) of the variety #math133##tex2html_wrap_inline4694# defined by <#1569#>

#math134#

#tex2html_wrap_indisplay4696# : = (p1,…, pδ-1, αδ,…, αr)∈#tex2html_wrap_indisplay4699##tex2html_wrap_indisplay4700#f1(#tex2html_wrap_indisplay4701#) = … = fρ+1(#tex2html_wrap_indisplay4702#) = 0≠g(#tex2html_wrap_indisplay4703#)

<#1569#> and the 0-dimensional radical ideal

#math135#

#tex2html_wrap_indisplay4707# : = #tex2html_wrap_indisplay4708#I(#tex2html_wrap_indisplay4711#) = #tex2html_wrap_indisplay4714# + (Z1 - p1,…, Zδ-1 - pδ-1).