 Saddle point

For a holomorphic phase $\varphi$, a saddle point is a critical point $z_0$ with ${\varphi }^{\prime }(z_0)=0$. At a simple saddle, ${\varphi }^{\prime \prime }(z_0)\ne 0$, and the quadratic Taylor term determines the local descent directions.

A saddle point is a point on the surface of a graph where the slope (or derivative) is zero in multiple dimensions, but is not a local extremum (i.e., not a local maximum or minimum). It occurs in both single-variable and multivariable calculus, although the characteristics can differ slightly based on the context.