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In this work we prove the local existence of absolute $p$-bases for regular algebras of finite type over $k$. Namely, consider a regular variety $Z$ over $k$. Kimura and Niitsuma proved that, for every $\\xi \\in Z$, the local ring $\\mathcal{O}_{Z,\\xi}$ has a $p$-basis over $\\mathcal{O}_{Z,\\xi}^p$. 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