Sign test of multivariable real polynomials

New theorems and procedures are presented for testing positivity and nonnegativity of a real polynomial in
n
real variables. The methods pertain to global as well as nonglobal or local sign tests for any closed intervals of the variables. A dimension growth is not required, and the methods depend on resolution of the solution of a set of
n
equations in
n
variables. Also provided is a simplified estimation of the computational effort involved in the present procedures, as compared to procedures where a dimension growth is required. Several illustrative examples are also provided.