An Ecumenic Notion of Solving

As the decomposition algorithms (Chapter~35) were reducing primary decomposition of multivariate ideals to factorization of univariate polynomials, Trinks' Algorithm (as most other solving algorithms) reduces multivariate zero-dimensional ideal solving, to univariate polynomial solving.

Most of these algorithms are 'ecumenic', in the sense that they can be applied to any computational model of L which allows, in <#623#>endlichvielen Schritten<#623#>,

to computationally perform the four operations in L,
and, for each univariate polynomial #math137#p(T)∈L[T], to 'solve' it, <#625#>id est<#625#> to produce the set #math138#{a∈L : p(a) = 0} of all the roots of p living in L,

and use these tools, given F, to 'solve' the zero-dimensional ideal #tex2html_wrap_inline3646# generated by F, <#628#>id est<#628#> to produce the set #math139##tex2html_wrap_inline3649#Z(#tex2html_wrap_inline3650#)∩L^r of all the roots of #tex2html_wrap_inline3652# with coordinates in L.

Trink's Algorithm (Figure~#GKA#632>) is a perfect instance of such `ecumenic' algorithms: for instance, setting #math140#L : = #tex2html_wrap_inline3655#R, it can be <#634#>verbatim<#634#> applied to a numerical analysis solver, or adapted in order to make use of Sturm Representation and Thom Codification of Algebraic Reals.

In the same way, Trink's Algorithm can be easily adapted in order to make use of Kronecker's (and Duval's) Model; obviously the resulting algorithm (Figure~#TAKM#639>) is a <#640#>verbatim<#640#> reformulation of the Zero-dimensional Prime Decomposition Algorithm discussed in Section~35.2. Such strict relation between 'solving' and decomposing, which was already stressed in Section~34.5, is just a simple consequence of Kronecker--Duval Philosophy.

**<#3662#>Figure<#3662#>:** <#3663#>Trinks' Algorithm in Kronecker's Model<#3663#>
#figure641#

All over this Part we will preserve this `ecumenic' approach to the notion of 'solving', as much as the persented solvers will allow to do so; naturally, the most strict solver presented here is an integralist 25 version of Kronecker--Duval Phylosophy.