Polynomial Cycles in Finitely Generated Domains.
All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective. SISI rings include Noetherian rings, Morita rings and almost maximal valuation rings ([V1]). In [F3] we raised the question of whether a polynomial ring R[x] over a SISI ring R is again SISI. In this paper we show this is not the case.
Let be a polynomial with integral coefficients. Shapiro showed that if the values of at infinitely many blocks of consecutive integers are of the form , where is a polynomial with integral coefficients, then for some polynomial . In this paper, we show that if the values of at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form where is an integer greater than 1, then for some polynomial .
Let be a field and . Let be a monomial ideal of and be monomials in . We prove that if form a filter-regular sequence on , then is pretty clean if and only if is pretty clean. Also, we show that if form a filter-regular sequence on , then Stanley’s conjecture is true for if and only if it is true for . Finally, we prove that if is a minimal set of generators for which form either a -sequence, proper sequence or strong -sequence (with respect to the reverse lexicographic...
Soit un corps commutatif. Chercher une série formelle vérifiant conduit naturellement à étudier l’application , étant une unité de l’algèbre , et à ramener les solutions à la forme , étant une suite de vérifiant les “identités multinomiales” :Après mise à l’écart par des lemmes combinatoires du cas caract (les solutions sont triviales), on caractérise de plusieurs manières les solutions. On peut les faire coïncider avec l’ensemble NW des suites de polynômes (ou séries génératrices...