If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominators of very high degrees. When the base field is $\mathbb{C}$, we give a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as of the largest modulus of the local exponents at all its singularities. As a consequence, if a differential operator $L$ has rational function coefficients over a number field, we get degree bounds for its monic right factors in terms of the degree, the order and the height of $L$, and of the degree of the number field.

中文翻译：

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线性微分算子右因子的显式界

如果具有有理函数系数的线性微分算子是可约的，则其因数可能具有非常高的分子和分母的系数。当基本字段是$\mathbb{C}$，我们根据原始算子的阶数和阶数以及所有奇异点处的局部指数的最大模数，为单项权利因子的阶数给出了一个明确的界限。结果，如果是微分算子$\mathrm{大号}$ 在数域上具有有理函数系数，我们根据其阶数，阶数和高度获得其单项权利因子的阶数边界 $\mathrm{大号}$，以及数字字段的度数。