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Let L(y) = 0 be a linear differential equation with rational functions as coefficients. To solve L(y) = 0 it is very helpful if the problem could be reduced to solving linear differential equations of lower order. One way is to compute a factorization of L, if L is reducible. Another way is to see if an operator L of order greater than 2 is a symmetric power of a second order operator. Maple contains implementations for both of these. The next step would be to see if L is a symmetric product of...

Let $F$ be a field whose characteristic is not $2$ and $K=F\left(t\right)$. We give a simple algorithm to find, given $a,b,c\in K^{*}$, a nontrivial solution in $K$ (if it exists) to the equation $a{X}^2+b{Y}^2+c{Z}^2=0$. The algorithm requires, in certain cases, the solution of a similar equation with coefficients in $F$; hence we obtain a recursive algorithm for solving diagonal conics over $\mathbb{Q}(t_1,\cdots ,t_n)$ (using existing algorithms for such equations over $\mathbb{Q}$) and over ${𝔽}_q(t_1,\cdots ,t_n)$.

Let $\mathbb{Q}\left(\alpha \right)$ and $\mathbb{Q}\left(\beta \right)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\mathbb{Q}\left(\beta \right)\to \mathbb{Q}\left(\alpha \right)$. The algorithm is particularly efficient if there is only one isomorphism.

We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.