By generalizing the concept of homogeneous polynomial and by adapting Cauchy's technique for obtaining bounds on the zeros of polynomials in one complex variable, the level surfaces of a real polynomial in $E^n$ are studied with respect to their intersection with certain curves, including all lines, passing through the origin. In addition, it is shown that the equipotential surface of any axisymmetric harmonic polynomial in $E^3$ is unbounded if and only if it is asymptotic to a finite union of cones each of which is parallel to a cone having the origin as its vertex. This paper extends results obtained by M. Marden and P. A. McCoy in 1976.