1.1 By R we denote an integral domain; then R[[x]] is the ring of formal power series over R in the variables x = (x1, . . . , xn). We write a series f(x) ∈ R[[x]] as f(x) = ∑ ai1···inx i1 1 · · ·xin n = ∑ aνx ν where ν = (ν1, . . . , νn) is a multi-index and x = x1 1 · · ·xνn n . In the sequel R is usually the finite field Fp of p elements or the ring of p-adic integers Zp. One says that f(x) ...

We refine necessary and sufficient conditions for the generating series of a weighted model quarter plane walk to be differentially algebraic. In addition, we give algorithms based on theory Mordell–Weil lattices, that, each model, yield polynomial weights determining this property associated series.

To study power series generated by Lindenmayer systems we de ne L algebraic systems and series over arbitrary commutative semirings. We establish closure and xed point properties of L algebraic series. We show how the framework of L algebraic series can be used to de ne D0L, 0L, E0L, DT0L, T0L and ET0L power series. We generalize for power series the classical result stating that D0L languages ...

1: After identification of the algebra of exponential generating series with the shuffle algebra of ordinary formal power series, the exponential map exp! : XK[[X]] −→ 1 +XK[[X]] for the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field K by a result going back to Hurwitz. The main result of this paper states that exp! (and its recipro...

GF(2)-grammars are a recently introduced grammar family that has some unusual algebraic properties and is closely connected to the of unambiguous grammars. By using method formal power series, we establish strong conditions necessary for subsets $$a_1^* a_2^* \cdots a_k^*$$ be described by GF(2)-grammar. further applying established results, settle long-standing open question proving inherent a...

We study multidimensional configurations (infinite words) and subshifts of low pattern complexity using tools of algebraic geometry. We express the configuration as a multivariate formal power series over integers and investigate the setup when there is a non-trivial annihilating polynomial: a non-zero polynomial whose formal product with the power series is zero. Such annihilator exists, for e...