In mathematics, the **Nakai conjecture** states that if *V* is a complex algebraic variety, such that its ring of differential operators is generated by the derivations it contains, then *V* is a smooth variety. This is the conjectural converse to a result of Alexander Grothendieck. It is known to be true for algebraic curves. The conjecture was proposed by the Japanese mathematician Yoshikazu Nakai.

A consequence would be the **Zariski-Lipman conjecture**, for a complex variety *V* with coordinate ring *R*: if the derivations of *R* are a free module over *R*, then *V* is smooth.

### Famous quotes containing the word conjecture:

“There is something fascinating about science. One gets such wholesale returns of *conjecture* out of such a trifling investment of fact.”

—Mark Twain [Samuel Langhorne Clemens] (1835–1910)