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Session S24 - Symbolic Computation: Theory, Algorithms and Applications

Tuesday, July 20, 16:00 ~ 16:25 UTC-3

Mahler residues and telescopers for rational functions

Carlos Arreche

The University of Texas at Dallas, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloake800777e2dce37b7139c91b46c1bf695').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addye800777e2dce37b7139c91b46c1bf695 = '&#97;rr&#101;ch&#101;' + '&#64;'; addye800777e2dce37b7139c91b46c1bf695 = addye800777e2dce37b7139c91b46c1bf695 + '&#117;td&#97;ll&#97;s' + '&#46;' + '&#101;d&#117;'; var addy_texte800777e2dce37b7139c91b46c1bf695 = '&#97;rr&#101;ch&#101;' + '&#64;' + '&#117;td&#97;ll&#97;s' + '&#46;' + '&#101;d&#117;';document.getElementById('cloake800777e2dce37b7139c91b46c1bf695').innerHTML += '<a ' + path + '\'' + prefix + ':' + addye800777e2dce37b7139c91b46c1bf695 + '\'>'+addy_texte800777e2dce37b7139c91b46c1bf695+'<\/a>';

We develop a notion of Mahler discrete residues for rational functions, with the desired property that a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ (where $p$ is an integer $\geq 2$) if and only if all of its Mahler discrete residues vanish. We also show how to apply the technology of Mahler discrete residues to creative telescoping problems. This work extends to the Mahler case the earlier analogous notions, properties, and applications of discrete residues (in the shift case) and $q$-discrete residues (in the $q$-difference case) developed by Chen and Singer.

Joint work with Yi Zhang (Xi'an Jiaotong-Liverpool University, China).