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...deal I of Section 1 is either (1) or (x , ) for some 2 K, so, together with the algorithm of [8] for the completely reducible cases, this yields a rational heuristic alternative toKovacic's algorithm =-=[5]-=-. 6 Timings and Comparisons We have implemented in the MAPLE computer algebra system the method described in Section 5 for computing the rational solutions of rst order Riccati equations. The heuristi...

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...ds and for rst order equations, the nonlinear systems produced by this approach are usually beyond the capacity of algebraic solvers. We propose in this paper a generalization of the series method of =-=[1]-=- to equations of type (1) that produces systems in n indeterminates regardless of the degree bounds. In the case of Riccati equations, this method is similar to the one in [10], except that we can cho...

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...2y 00 +(2a 2 1 +4a0a2 + a 0 1a2 , a1a 0 2)y 0 +(4a0a1 +2a 0 0a2 , 2a0a 0 2)y =0 does not have nontrivial rational solutions in K(x), then it is reducible if and only if it is not completely reducible =-=[8]-=-, so if the equation (16) has a solution y 2 K(x), where K is the algebraic closure of K, then it has a unique such solution, which must in fact be in K(x). This means that when computing the power se...

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...a0;:::;an,1)(x , a) k . If the degrees of the numerator and denominator of y are bounded by L and M respectively, then we can compute y from its power series by computing its [L=M] Pade approximation =-=[2]-=-. Of course we do not know a0;:::;an,1 a priori so we try to nd algebraic equations whose solutions give candidate values for a0;:::;an,1. Let N 0beaninteger. Using Pade approximation we can nd polyno...

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...lgebraic extension of K. For any irreducible factor p of an in E[x], one can compute an upper bound mp for the power of p that appears in the denominator of y. This is classical and goes back to Beke =-=[6,7]-=-. It is in fact possible to compute mp for all the factors of an and D = Q pjan pmp without factoring an [3]. In a similar way, we can compute an upper bound m1 for the degree of the polynomial part o...

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...lgebraic extension of K. For any irreducible factor p of an in E[x], one can compute an upper bound mp for the power of p that appears in the denominator of y. This is classical and goes back to Beke =-=[6,7]-=-. It is in fact possible to compute mp for all the factors of an and D = Q pjan pmp without factoring an [3]. In a similar way, we can compute an upper bound m1 for the degree of the polynomial part o...

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... of the series method of [1] to equations of type (1) that produces systems in n indeterminates regardless of the degree bounds. In the case of Riccati equations, this method is similar to the one in =-=[10]-=-, except that we can choose to expand at an ordinary point of the equation rather than at a singularity, which yields better performances when the singularities are algebraic rather than rational numb...

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... rational solutions of rst order Riccati equations. The heuristic bound on the number of extra singularities is computed from modular factorizations, which are done via a call to the bernina 1 server =-=[4]-=-. We have used the following examples to test the e ectiveness of our method: L1 = d dx + x2 +2 x 7 , 3 L2 = d dx + x6 , 2 x13 +3x11 , 2x2 , 3 L3 = d dx + 3x2 , 2 x5 +3x3 , 2x2 , 2 L4 = d 1 + dx x5 , ...

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...yielding a series P k0 T k (x \Gamma a) k , and then use P N k=0 T k (x \Gamma a) k as candidate solution, in which case I m is the ideal (TN+1 ; : : : ; TN+m ). Together with the bounding methods of =-=[3]-=-, this yields an algorithm for finding the rational solutions with no extra poles (i.e. whose poles are among the singularities of the equation) of generalized Riccati equations of the type (2). A met...

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...her than at a singularity, which yields better performances when the singularities are algebraic rather than rational numbers. In addition, when coupling this method with a modular heuristic based on =-=[9]-=- for guessing the number of extra singularities, our method yields a rational heuristic factorizer for second order linear ordinary differential operators, which is particularly efficient for non comp...

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... of the series method of [1] to equations of type (1) that produces systems in n indeterminates regardless of the degree bounds. In the case of Riccati equations, this method is similar to the one in =-=[10]-=-, except that we can choose to expand at an ordinary point of the equation rather than at a singularity, which yields better performances when the singularities are algebraic rather than rational numb...

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...her than at a singularity, which yields better performances when the singularities are algebraic rather than rational numbers. In addition, when coupling this method with a modular heuristic based on =-=[9]-=- for guessing the number of extra singularities, our method yields a rational heuristic factorizer for second order linear ordinary di erential operators, which is particularly e cient for non complet...

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...k 0 Sk(x , a) k by d, yielding a series P k 0 Tk(x , a) k , and then use PN k=0 Tk(x , a) k as candidate solution, in which case Im is the ideal (TN+1;:::;TN+m). Together with the bounding methods of =-=[3]-=-, this yields an algorithm for nding the rational solutions with no extra poles (i.e. whose poles are among the singularities of the equation) of generalized Riccati equations of the type (2). A metho...