Mourrain: Proving Cardinal's Conjecture
Let us use the same notation as in the last sections; in particular we have
#math771##tex2html_wrap_inline13422# : = {a1,…, aδ}, #math772##tex2html_wrap_inline13424# : = {b1,…, bδ}
#math773#A = #tex2html_wrap_inline13426#(#tex2html_wrap_inline13427#), #math774#B = #tex2html_wrap_inline13429#(#tex2html_wrap_inline13430#)
and we set
#math775##tex2html_wrap_inline13432# : = M0-1Mp : = #tex2html_wrap_inline13433#mji(p)#tex2html_wrap_inline13434#.
#Proposition4294#
#proof4340#
#Corollary4411#
With a slight abuse of notation we will also denote #math776##tex2html_wrap_inline13436# the map
#math777#
#tex2html_wrap_indisplay13438# : A→A, ai #tex2html_wrap_indisplay13439# #tex2html_wrap_indisplay13440#mml(p)al
which corresponds to the multiplication by Xp modulo #math778##tex2html_wrap_inline13443#.
Since these operations commute, for each
#math779#f (X1,…, Xn)∈#tex2html_wrap_inline13445#P we define
#math780#
f (#tex2html_wrap_indisplay13447#) : = f (#tex2html_wrap_indisplay13448#,…,#tex2html_wrap_indisplay13449#) : A→A
and
#math781#N(f )= f (#tex2html_wrap_inline13451#)(1) so that N is a map #math782#N : #tex2html_wrap_inline13454#P→A.
#Proposition4428#
#proof4438#
#Proposition4455#
#proof4467#
#Theorem4527#
#proof4536#