Rational representation of modular numbers (original) (raw)

1999, Matemática Contemporânea

\Ve fincl conditlons, restricting the size of the fractions, and present algorithms to obtaln a rational number from a set of resiclues modulo relatively prime integers. \Ve also discuss the na.t.ure anel the num her of solutions for the rat.iona.l represe11tat.ion, introducing condit.ions for the cxü;tcncc and for Lhe uniqucnc:::;:::;. Resumo Itcstrlnglndo o tamanho das fra<;ôcs, concliçôcs são estabelecidas c algoritmos são apresentados para q uc um n úmcro racional seja obtido a. partir de um conjunto de resíduos módulo inteiros que sào primos entre si. Ta.m bP m sào d iscutidns a. na.tu re7.a e o n tÍ mero de soluçôes para, a reprel'enl.ar:<io racion ai, estabelecendo cond ir:óel' para a exisLênci a e 11 nicidade. 1. Introd uction ln Symbolic and Algebraic Computation, a wide range of problems have been solved very effi.ciently via, a modulaT approach. Many problems over the integers are mapJHcd on1o a prime field a.nd once !lu-: ima.ge oi' Ute soluLion is known, i1 is recovered Lo a 1rue solu1ion for tlw original problent, by meam of Lhe chinese remainder or HenselJ;.-adic lif1ing algoritlnrt. Represent.a1ive examples of this approach include polynomial factorization algorithms (see, for example, [3]), pol:ynomial GCD computations (scc, for instancc, [3]) and Grobncr basis computation algorithms [R]. Smne of" Lltese problems would be more dficienLly solved over t.l1e ra1ionals, and other prohlems arise more na.tmally in this field. Ca.n we de~ign an effir:ient