Nonsingular point of an algebraic curve ($F_x(P)\ne 0\text{ or }{F}_y(P)\ne 0$)

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.

For a projective plane curve $C=V(F)$ and a nonsingular projective point $p$, the tangent is the line $F_x(p)x+F_y(p)y+F_z(p)z=0$. The coefficients are not all zero, and the Euler theorem for homogeneous functions makes this line pass through $p$. On an affine chart this is precisely the first-order part of a local defining equation, so the definition agrees with the usual tangent line and is invariant under projective transformations.