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Cyclically valued rings and formal power series

Gérard Leloup (2007)

Annales mathématiques Blaise Pascal

Rings of formal power series k [ [ C ] ] with exponents in a cyclically ordered group C were defined in [2]. Now, there exists a “valuation” on k [ [ C ] ] : for every σ in k [ [ C ] ] and c in C , we let v ( c , σ ) be the first element of the support of σ which is greater than or equal to c . Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in k [ [ C ] ] . We prove that a cyclically valued ring is a subring of a power series ring k [ [ C , θ ] ] with...

Degree estimate for subalgebras generated by two elements

Leonid Makar-Limanov, Jie-Tai Yu (2008)

Journal of the European Mathematical Society

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, which plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong...

Diagonalization and rationalization of algebraic Laurent series

Boris Adamczewski, Jason P. Bell (2013)

Annales scientifiques de l'École Normale Supérieure

We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime p the reduction modulo p of the diagonal of a multivariate algebraic power series f with integer coefficients is an algebraic power series of degree at most p A and height at most A p A , where A is an effective constant that only depends on...

Division dans l'anneau des séries formelles à croissance contrôlée. Applications

Augustin Mouze (2001)

Studia Mathematica

We consider subrings A of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove a Weierstrass-Hironaka division theorem for such subrings. Moreover, given an ideal ℐ of A and a series f in A we prove the existence in A of a unique remainder r modulo ℐ. As a consequence, we get a new proof of the noetherianity of A.

Equations in the Hadamard ring of rational functions

Andrea Ferretti, Umberto Zannier (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let K be a number field. It is well known that the set of recurrencesequences with entries in K is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume { a n } is a recurrence sequence and suppose that all the a n have a d th root in the field K ; then (after...

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