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

Abstract

We prove that homogeneous symmetric polynomial inequalities of degree p ∈ { 4 , 5 } in n positive 1 1 Which will mean ⩾0, according to the terminology of ordered vector spaces. 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 ( H p [ n ] , ⪯ ) and ( H p [ 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.