Abstract
Let L be a linear difference operator with polynomial coefficients. We consider singularities of L that correspond to roots of the trailing (resp. leading) coefficient of L. We prove that one can effectively construct a left multiple with polynomial coefficients 
of L such that every singularity of is a singularity of L that is not apparent. As a consequence, if all singularities of L are apparent, then L has a left multiple whose trailing and leading coefficients equal 1.
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References
Abramov, S.A.: Rational solutions of linear difference and q-difference equations with polynomial coefficients. Programming and Comput. Software (1995) 21, 273–278. (Translated from Programmirovanie (1995) 21, 3–11.)
Abramov, S.A., van Hoeij, M.: Desingularization of linear difference operators with polynomial coefficients. Proc. ISSAC'99, Vancouver (1999) pp. 269–275
Abramov, S.A., van Hoeij, M.: Set of poles of solutions of linear difference equations with polynomial coefficients. J. Comput. Math. Math. Phys. (2003) 43(1), 57–62 (Translated from Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki (2003) 43(1), 60–65.)
Barkatou, M.A.: Contribution à l'étude des équations différentielles et aux différences dans le champ complexe, PhD Thesis (1989) INPG, Grenoble France
van Hoeij, M.: Formal solutions and factorization of differential operators with power series coefficients. J. Symb. Comput. 24, 1–30 (1997)
van Hoeij, M.: Finite singularities and hypergeometric solutions of linear recurrence equations. J. Pure Appl. Algebra 139, 109–131 (1999)
Immink, G.K.: On the relation between linear difference and differential equations with polynomial coefficients. Math. Nachr. 200, 59–76 (1999)
Ince, E.L.: Ordinary differential equations. New York: Dover, 1944
Man, Y.K., Wright, F.G.: Fast polynomial dispersion computation and its application to indefinite summation. Proc. ISSAC'94, Oxford (1994), pp. 175–180
Mitichkina, A.M.: On an implementation of desingularization of linear recurrence operators with polynomial coefficients. Proc. of the international workshop ``Computer Algebra and its application to Physics'' (CAAP'2001), Russia, Dubna (2001), pp. 212–220
Petkovsek, M.: Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comput. 14, 243–264 (1992)
Ramis, J.-P.: Etude des solutions méromorphes des équations aux différences linéaires algébriques, manuscript
Tsai, H.: Weyl Closure of linear differential operator. J. Symb. Comput. 29, 747–775 (2000)
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Abramov, S., Barkatou, M. & van Hoeij, M. Apparent singularities of linear difference equations with polynomial coefficients. AAECC 17, 117–133 (2006). https://doi.org/10.1007/s00200-005-0193-9
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DOI: https://doi.org/10.1007/s00200-005-0193-9