  In the paper we investigate locally symmetric polynomial metrics in special
cases of Riemannian and Finslerian surfaces. The Riemannian case will be
presented by a collection of basic results (regularity of second root metrics)
and formulas up to Gauss curvature. In case of Finslerian surfaces we formulate
necessary and sufficient conditions for a locally symmetric fourth root metric
in 2D to be positive definite. They are given in terms of the coefficients of
the polynomial metric to make checking the positive definiteness as simple and
direct as possible. Explicit examples are also presented. The situation is more
complicated in case of spaces of dimension more than two. Some necessary
conditions and an explicit example are given for a positive definite locally
symmetric polynomial metric in 3D. Computations are supported by the MAPLE
mathematics software (LinearAlgebra).
