= Nonsingular point of an algebraic curve
{title2=$F_x(P)\ne0\text{ or }F_y(P)\ne0$}
For an affine plane <algebraic curve> $F(x,y)=0$, a point over an <algebraic closure> is nonsingular when the displayed condition holds. Its <Zariski tangent space> has dimension one, and a local coordinate can be chosen in the direction of a nonzero <partial derivative>. A <smooth algebraic curve> has all points nonsingular. Over a complete <local field>, the <Hensel lemma> lifts a nonsingular residue point by solving for a coordinate with unit <partial derivative>.
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