70 research outputs found
A Residue Theorem for Malcev-Neumann Series
In this paper, we establish a residue theorem for Malcev-Neumann series that
requires few constraints, and includes previously known combinatorial residue
theorems as special cases. Our residue theorem identifies the residues of two
formal series that are related by a change of variables. We obtain simple
conditions for when a change of variables is possible, and find that the two
related formal series in fact belong to two different fields of Malcev-Neumann
series. The multivariate Lagrange inversion formula is easily derived and
Dyson's conjecture is given a new proof and generalized.Comment: 22 pages, extensive revisio
Degree estimate for commutators
Let K be a free associative algebra over a field K of characteristic 0 and
let each of the noncommuting polynomials f,g generate its centralizer in K.
Assume that the leading homogeneous components of f and g are algebraically
dependent with degrees which do not divide each other. We give a counterexample
to the recent conjecture of Jie-Tai Yu that deg([f,g])=deg(fg-gf) >
min{deg(f),deg(g)}. Our example satisfies deg(g)/2 < deg([f,g]) < deg(g) <
deg(f) and deg([f,g]) can be made as close to deg(g)/2 as we want. We obtain
also a counterexample to another related conjecture of Makar-Limanov and
Jie-Tai Yu stated in terms of Malcev - Neumann formal power series. These
counterexamples are found using the description of the free algebra K
considered as a bimodule of K[u] where u is a monomial which is not a power of
another monomial and then solving the equation [u^m,s]=[u^n,r] with unknowns
r,s in K.Comment: 18 page
A Fast Algorithm for MacMahon's Partition Analysis
This paper deals with evaluating constant terms of a special class of
rational functions, the Elliott-rational functions. The constant term of such a
function can be read off immediately from its partial fraction decomposition.
We combine the theory of iterated Laurent series and a new algorithm for
partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega
calculus, which (partially) avoids the "run-time explosion" problem when
eliminating several variables. We discuss the efficiency of our algorithm by
investigating problems studied by Andrews and his coauthors; our running time
is much less than that of their Omega package.Comment: 22 page
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