Extrema of a real polynomial

In this paper, we investigate critical point and extrema structure of a multivariate real polynomial. We classify critical surfaces of a real polynomial f into three classes: repeated, intersected and primal critical surfaces. These different critical surfaces are defined by some essential factors of f, where an essential factor of f means a polynomial factor of f - c0, for some constant c0. We show that the degree sum of repeated critical surfaces is at most d - 1, where d is the degree of f. When a real polynomial f has only two variables, we give the minimum upper bound for the number of other isolated critical points even when there are nondegenerate critical curves, and the minimum upper bound of isolated local extrema even when there are saddle curves. We show that a normal polynomial has no odd degree essential factors, and all of its even degree essential factors are normal polynomials, up to a sign change. We show that if a normal quartic polynomial f has a normal quadratic essential factor, a global minimum of f can be either easily found, or located within the interior(s) of one or two ellipsoids. We also show that a normal quartic polynomial can have at most one local maximum.