On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity criteria for degrees 4 and 5

We prove that homogeneous symmetric polynomial inequalities of degree p∈{4,5} in n positive1 variables can be algorithmically tested, on a finite set depending on the given inequality (Theorem 13); the test-set can be obtained by solving a finite number of equations of degree not exceeding p−2. Section 3 discusses the structure of the ordered vector spaces (Hp[n],⪯) and (Hp[n],⋞). In Section 4, positivity criteria for degrees 4 and 5 are stated and proved. The main results are Theorems 10-14. Part III of this work will be concerned with the construction of extremal homogeneous symmetric polynomials (best inequalities) of degree 4 in n positive variables.