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The fundamental constituents of iteration digraphs of finite commutative rings

Jizhu Nan, Yangjiang Wei, Gaohua Tang (2014)

Czechoslovak Mathematical Journal

For a finite commutative ring R and a positive integer k ⩾ 2 , we construct an iteration digraph G ( R , k ) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a k . Let R = R 1 ⊕ ... ⊕ R s , where s > 1 and R i is a finite commutative local ring for i ∈ { 1 , ... , s } . Let N be a subset of { R 1 , ⋯ , R s } (it is possible that N is the empty set ∅ ). We define the fundamental constituents G N * ( R , k ) of G ( R , k ) induced by the vertices which are of the form { ( a 1 , ⋯ , a s ) ∈ R : a i ∈ D ( R i ) if R i ∈ N , otherwise a i ∈ U ( R i ) , i = 1 , ... , s } , where U ( R ) denotes the unit group of R and D ( R ) denotes the zero-divisor set of R . We investigate...

The fundamental theorem of prehomogeneous vector spaces modulo p m (With an appendix by F. Sato)

Raf Cluckers, Adriaan Herremans (2007)

Bulletin de la Société Mathématique de France

For a number field K with ring of integers 𝒪 K , we prove an analogue over finite rings of the form 𝒪 K / 𝒫 m of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where 𝒫 is a big enough prime ideal of 𝒪 K and m > 1 . In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113(1998), 237–346] to the functional equation of L -functions...

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