= Distributional regularity of a constant-coefficient ordinary differential equation
{title2=$P(D)v=0\Longrightarrow v\in C^\omega$}
For a nonzero constant-coefficient polynomial differential operator in one variable, every homogeneous <distribution> solution is a <classical solution>, indeed analytic. Factor the <polynomial> into powers of distinct <characteristic roots of a constant-coefficient differential equation>. The <kernel decomposition for coprime polynomials> reduces the equation to $(D-\lambda)^m v=0$, and <multiplication of a distribution by a smooth function> reduces this to $D^m(e^{-\lambda x}v)=0$. The fact that <a distribution with zero derivative is constant>, iterated, makes the latter distribution a <polynomial> of degree less than $m$.
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