scholarly journals Noncommutative rational Pólya series

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Jason Bell ◽  
Daniel Smertnig
Keyword(s):  
Number Theory ◽  
Power Series ◽  
Formal Power Series ◽  
Finite Automaton ◽  
Automata Theory ◽  
Finitely Generated ◽  
Noncommutative Algebra ◽  
Rational Series ◽  
Unit Equations ◽  
Formal Power

AbstractA (noncommutative) Pólya series over a field K is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $$K^\times $$ K × . We show that rational Pólya series are unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a Pólya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton.

scholarly journals The Poincaré series of the module of derivations of some monomial rings

2008 ◽  
Vol 102 (1) ◽  
pp. 5
Author(s):  
V. Micale
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Poincaré Series ◽  
Finitely Generated ◽  
Poincare Series ◽  
Formal Power

Let $R$ be a quasi-homogeneous $k$-algebra and $M$ be a finitely generated graded $R$-module. The formal power series $\sum_{i}\dim_{k}(\mathrm{tor}_{i}^R(k,M)z^i$ is called the Poincaré series of $M$ and it is denoted by $P_{M}^R(z)$. We remark that the Poincaré series of the module of derivations of a monomial ring is rational and determine it in some cases.

scholarly journals Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem

2005 ◽  
Vol 2005 (23) ◽  
pp. 3767-3780
Author(s):  
L. Haddad ◽  
C. Helou ◽  
J. Pihko
Keyword(s):  
Number Theory ◽  
Power Series ◽  
Formal Power Series ◽  
Positive Constant ◽  
Additive Number Theory ◽  
Natural Numbers ◽  
Additive Number ◽  
Bounded Below ◽  
Formal Power

We consider and study formal power series, that we call supported series, with real coefficients which are either zero or bounded below by some positive constant. The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. It is this analogy that we pursue, thus establishing many properties and giving equivalent statements to the well-known Erdös-Turán conjectures in terms of supported series and extending to them a version of Erdös-Fuchs theorem.

scholarly journals NON-COMMUTATIVE CHARACTERISTIC POLYNOMIALS AND COHN LOCALIZATION

2001 ◽  
Vol 64 (1) ◽  
pp. 13-28 ◽  
Author(s):  
DESMOND SHEIHAM
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Equivalence Class ◽  
Characteristic Polynomial ◽  
Constant Coefficient ◽  
Characteristic Polynomials ◽  
Invertible Matrix ◽  
Finitely Generated ◽  
Formal Power

Almkvist proved that for a commutative ring A the characteristic polynomial of an endomorphism α : P → P of a finitely generated projective A-module determines (P, α) up to extensions. For a non-commutative ring A the generalized characteristic polynomial of an endomorphism of an endomorphism α : P → P of a finitely generated projective A-module is defined to be the Whitehead torsion [1 − xα] ∈ K1(A[[x]]), which is an equivalence class of formal power series with constant coefficient 1.The paper gives an example of a non-commutative ring A and an endomorphism α : P → P for which the generalized characteristic polynomial does not determine (P, α) up to extensions. The phenomenon is traced back to the non-injectivity of the natural map [sum ]−1A[x] → A[[x]] where [sum ]−1A[x] is the Cohn localization of A[x] inverting the set [sum ] of matrices in A[x] sent to an invertible matrix by A[x] → A;x [map ] 0.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

Sign in / Sign up

Export Citation Format

Share Document