Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFactorisation theorems for generalised power series
50
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Fields of generalised power series (or Hahn fields), with coefficients in a field and exponents in a divisible ordered abelian group, are a fundamental tool in the study of valued and ordered fields and asymptotic expansions. The subring of the series with non-positive exponents appear naturally when discussing exponentiation, as done in transseries, or integer parts. A notable example is the ring of omnific integers inside the field of Conways surreal numbers. In general, the elements of such subrings do not have factorisations into irreducibles. In the context of omnific integers, Conway conjectured in 1976 that certain series are irreducible (proved by Berarducci in 2000), and that any two factorisations of a given series share a common refinement. Here we prove a factorisation theorem for the ring of series with non-positive real exponents: every series is shown to be a product of irreducible series with infinite support and a factor with finite support which is unique up to constants. From this, we shall deduce a general factorisation theorem for series with exponents in an arbitrary divisible ordered abelian group, including omnific integers as a special case. We also obtain new irreducibility and primality criteria. To obtain the result, we prove that a new ordinal-valued function, which we call degree, is a valuation on the ring of generalised power series with real exponents, and we formulate some structure results on the associated RV monoid.
rate research
Read More
A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. A fundamental result of Berarducci ensures the existence of irreducible series in the subring $K((G^{leq 0}))$ of $K((G))$ consisting of the generalised power series with non-positive exponents. It is an open question whether the factorisations of a series in such subring have common refinements, and whether the factorisation becomes unique after taking the quotient by the ideal generated by the non-constant monomials. In this paper, we provide a new class of irreducibles and prove some further cases of uniqueness of the factorisation.
A theorem of alternatives provides a reduction of validity in a substructural logic to validity in its multiplicative fragment. Notable examples include a theorem of Arnon Avron that reduces the validity of a disjunction of multiplicative formulas in the R-mingle logic RM to the validity of a linear combination of these formulas, and Gordans theorem for solutions of linear systems over the real numbers, that yields an analogous reduction for validity in Abelian logic A. In this paper, general conditions are provided for axiomatic extensions of involutive uninorm logic without additive constants to admit a theorem of alternatives. It is also shown that a theorem of alternatives for a logic can be used to establish (uniform) deductive interpolation and completeness with respect to a class of dense totally ordered residuated lattices.
In this paper, we give new proofs of the celebrated Andreka-Resek-Thompson representability results of certain axiomatized cylindric-like algebras. Such representability results provide completeness theorems for variants of first order logic, that can also be viewed as multi-modal logics. The proofs herein are combinatorial and we also use some techniques from game theory.
We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $mathbb{E}_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence relation and $cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $Ein {cal F}$ or $F$ is Borel reducible to $E$, then $cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
Elements of the Riordan group $cal R$ over a field $mathbb F$ of characteristic zero are infinite lower triangular matrices which are defined in terms of pairs of formal power series. We wish to bring to the forefront, as a tool in the theory of Riordan groups, the use of multiplicative roots $a(x)^frac{1}{n}$ of elements $a(x)$ in the ring of formal power series over $mathbb F$ . Using roots, we give a Normal Form for non-constant formal power series, we prove a surprising simple Composition-Cancellation Theorem and apply this to show that, for a major class of Riordan elements (i.e., for non-constant $g(x)$ and appropriate $F(x)$), only one of the two basic conditions for checking that $big(g(x), , F(x)big)$ has order $n$ in the group $cal R$ actually needs to be checked. Using all this, our main result is to generalize C. Marshall [Congressus Numerantium, 229 (2017), 343-351] and prove: Given non-constant $g(x)$ satisfying necessary conditions, there exists a unique $F(x)$, given by an explicit formula, such that $big(g(x), , F(x)big)$ is an involution in $cal R$. Finally, as examples, we apply this theorem to ``aerated series $h(x) = g(x^q), q text{odd}$, to find the unique $K(x)$ such that $big(h(x), K(x)big)$ is an involution.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
