Newton map in optimization ($N(p)=p-H(p{)}^{-1}\nabla f(p)$)

For a twice continuously differentiable real function, the Newton map sends an input $p$ to the next iterate of Newton's method in optimization. It is defined where the Hessian matrix $H(p)$ is invertible. Iterating this map seeks a zero of a function of the gradient; additional information is needed to determine whether a resulting stationary point is a local minimum.