Footnotes

... perfect: <#1062#>While the techniques discussed here apply in this general setting we are mainly thinking of the case #math54#k = #tex2html_wrap_inline3256#Q,#tex2html_wrap_inline3257# : = #tex2html_wrap_inline3258#C; on the other side technically we need to (and we can) solve over #math55##tex2html_wrap_inline3260#Q(V₁,…, V_d).<#1062#>
... ideal: <#1063#>All over the book I will use the notation #math59##tex2html_wrap_inline3280#I(F)⊂#tex2html_wrap_inline3281#R in order to denote the ideal generated by the basis F in the ring #tex2html_wrap_inline3284#R; when there is no ambiguity #tex2html_wrap_inline3286#R will be not specified.<#1063#>
... ordering: <#1065#>Recall that (cf. Definition~<#51#>22.1.2<#51#>) a well-ordering ;SPMlt; on #tex2html_wrap_inline3323#T will be called a term ordering if it is a <#53#>semigroup ordering<#53#>, <#54#>id est<#54#>
#math67#
t₁ ;SPMlt; t₂#tex2html_wrap_indisplay3325#tt₁ ;SPMlt; tt₂,#tex2html_wrap_indisplay3326#t, t₁, t₂∈#tex2html_wrap_indisplay3327#T.
<#1065#>
...basis: <#1069#>The <#342#>leading polynomial<#342#> (page~#LePo#343>) $(g_i)$ is indicated in <#344#>bold<#344#>.<#1069#>
... solver: <#1070#>Of course, such a statement must be taken <#635#>cum grano salis<#635#>: it forgets the ill-conditioning problem, which requires at least some suitable pre-processing before applying the Algorithm.<#1070#>
... Reals: <#1071#>Chapter~13 dicusses both such representations and the techniques needed in order to solve the required polynomials
#math141#
p(Z_i) : = h(a₁,…, a_i-1, Z_i), h∈#tex2html_wrap_indisplay3659#Q[Z₁,…, Z_i],
where each #math142#a_i∈#tex2html_wrap_inline3661#R is given by such representation.<#1071#>
...case: <#1072#>Unlike Gröbner's argument which was not intended as an effictive solver and was turn into such by Trinks, Kronecker's argument was an effective solver.
The restriction to the zero-dimensional case is done here to simplify the argument but is <#785#>not<#785#> required by Kronecker's solver which in fact applies also to non-unmixed ideals.
The details on the general case are discussed in Sections~20.3 and 20.4.<#1072#>
...version: <#1073#>Delassus E., <#809#>Sur les systèmes algébriques et leurs relations avec certains systèmes d'́equations aux dérivées partielles<#809#>. Ann. Éc. Norm. $3^e$ série <#810#>14<#810#> (1897) 21--44<#1073#>
...Robinson: <#1074#>Gunther, N. <#811#>Sur les caractéristiques des systémes d'equations aux dérivées partialles<#811#>, C.R. Acad. Sci. Paris <#812#>156<#812#> (1913), 1147--1150 and Robinson, L.B. <#813#>Sur les systémes d'équations aux dérivées partialles<#813#> C.R. Acad. Sci. Paris <#814#>157<#814#> (1913), 106--108<#1074#>
...Gunther: <#1075#>Gunther, N. <#815#>Sur la forme canonique des systèmes d'équations homogènes<#815#> (in russian) [Journal de l'Institut des Ponts et Chaussées de Russie] Izdanie Inst. Inz̆. Putej Soobs̆c̆enija Imp. Al. I. <#816#>84<#816#> (1913) and Gunther, N. <#817#>Sur la forme canonique des équations algébriques<#817#>, C.R. Acad. Sci. Paris <#818#>157<#818#> (1913), 577--80 .<#1075#>
...and: <#1077#>the notation $<#826#>J<#826#>_d$ denotes here the set of the homogeneous members of $<#827#>J<#827#>$ of degree $d$ and must not be indentify with the previous notation where $<#828#>J<#828#>_i$ denotes the members of $<#829#>J<#829#>$ depending only on the first $j$ variables.<#1077#>
...value: <#1079#>We can set $D := {(g) : g&isin#in;G}$ where $G$ is a Gröbner basis of $<#860#>J<#860#>$ wrt $&pr#prec;$ but the existence can be easily derived (as for the finiteness of Gröbner bases) by Hilbert's Nullstellensatz and this is the approach used by Gunther.<#1079#>
...satisfies: <#1080#>This is a direct consequence of Corollary~37.2.8 applied to the set $<#871#>T<#871#>_&pr#prec;(f) = <#872#>L<#872#>_;SPMlt;(f)$ and to the (deg)-lex ordering $&pr#prec;$ induced by $Z_r &pr#prec;...&pr#prec;Z_2 &pr#prec;Z_1$.
Delassus' mistake is to assume that $<#873#>T<#873#>_&pr#prec;(<#874#>J<#874#>_d) = <#875#>L<#875#>_;SPMlt;(<#876#>J<#876#>_d)$ is the set $<#877#>L<#877#>(d)$ consists of the first $#<#878#>L<#878#>_;SPMlt;(<#879#>J<#879#>_d)$ terms w.r.t. the (deg)-revlex $;SPMlt;$ induced by $Z_1 ;SPMlt; Z_2 ;SPMlt; ...;SPMlt; Z_r$ which tantamount to the last $#<#880#>T<#880#>_&pr#prec;(<#881#>J<#881#>_d)$ terms w.r.t. the (deg)-lex ordering $&pr#prec;$ induced by $Z_r &pr#prec;...&pr#prec;Z_2 &pr#prec;Z_1$.
In its Lemma (<#882#>cf.<#882#> Section~23.3) Macaulay was considering the same set $<#883#>L<#883#>(d), #<#884#>L<#884#>(d) = #<#885#>L<#885#>_;SPMlt;(<#886#>J<#886#>_d)$ as Delassus and presented it, as Delassus, in terms of the (deg)-revlex ordering $;SPMlt;$ and not in terms of the (deg)-lex ordering $&pr#prec;$. <#1080#>
...Z_r]: <#972#>As a consequence of the elimination property of the lex ordering $&pr#prec;$ which is explicitly stated by Gunther.<#972#>
...and: <#1024#>Again a consequence of the elimination property of the lex ordering $&pr#prec;$ explicitly stated by Gunther.<#1024#>
...Gunther: <#1092#>It begins by considering a substitution
#math151#
Z₁ = U₁, Z₂ = AU₁ + U₂, Z₃ = U₃,…, Z_r = U_r,
where $A$ is a variable, and the corresponding equations in $K[A][U_1,...,U_r]$ which <#1059#>sont de fonction holomorphes de $A$ dans la voisinage de $A = 0$.<#1059#> <#1092#>