3,439 research outputs found
On the Invariance of Residues of Feynman Graphs
We use simple iterated one-loop graphs in massless Yukawa theory and QED to
pose the following question: what are the symmetries of the residues of a graph
under a permutation of places to insert subdivergences. The investigation
confirms partial invariance of the residue under such permutations: the highest
weight transcendental is invariant under such a permutation. For QED this
result is gauge invariant, ie the permutation invariance holds for any gauge.
Computations are done making use of the Hopf algebra structure of graphs and
employing GiNaC to automate the calculations.Comment: 24 pages, latex generated figures. Minor changes in revised versio
Decomposition of Algebraic Functions
AbstractFunctional decomposition—whether a functionf(x) can be written as a composition of functionsg(h(x)) in a non-trivial way—is an important primitive in symbolic computation systems. The problem of univariate polynomial decomposition was shown to have an efficient solution by Kozen and Landau (1989). Dickerson (1987) and Gathen (1990a) gave algorithms for certain multivariate cases. Zippel (1991) showed how to decompose rational functions. In this paper, we address the issue of decomposition of algebraic functions. We show that the problem is related to univariate resultants in algebraic function fields, and in fact can be reformulated as a problem ofresultant decomposition. We characterize all decompositions of a given algebraic function up to isomorphism, and give an exponential time algorithm for finding a non-trivial one if it exists. The algorithm involves genus calculations and constructing transcendental generators of fields of genus zero
PolyLogTools - Polylogs for the masses
We review recent developments in the study of multiple polylogarithms,
including the Hopf algebra of the multiple polylogarithms and the symbol map,
as well as the construction of single valued multiple polylogarithms and
discuss an algorithm for finding fibration bases. We document how these
algorithms are implemented in the Mathematica package PolyLogTools and show how
it can be used to study the coproduct structure of polylogarithmic expressions
and how to compute iterated parametric integrals over polylogarithmic
expressions that show up in Feynman integal computations at low loop orders.Comment: Package URL: https://gitlab.com/pltteam/pl
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