We prove that homogeneous symmetric polynomial inequalities of degree p ∈ {4, 5} in n positive 1 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 [n] p , ) and (H [n] p , ). 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.  2004 Elsevier Inc. All rights reserved.

We show that positivity on R n + and on R n of real symmetric polynomials of degree at most p in n ≥ 2 variables is solvable by algorithms running in poly(n) time. For real symmetric quartics, we find explicit discriminants and related Maple algorithms running in lin(n) time.

We prove strengthened and unified forms of vector-valued versions of the Stone-Weierstrass theorem. This is possible by using an appropriate factorization of a topological space, instead of the traditional localizability. Our main Theorem 7 generalizes and unifies number of known results. Applications from the last section include new versions in the scalar case, as well as simultaneous approximation and interpolation under additional constraints.

In this paper, we give a uniqueness theorem concerning a thermomechanical model which describes the behavior of shape memory alloys and takes into account the nonisothermal character of the phase transformations, as well as the existence of the intrinsic dissipation.

In this paper, which is a continuation of [V. Timofte, On the positivity of symmetric polynomial functions. Part I: General results, J. Math. Anal. Appl. 284 (2003) 174-190] and [V. Timofte, On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity criteria for degrees 4 and 5, J. Math. Anal. Appl., in press], we study properties of extremal polynomials of degree 4, and we give the construction of some of them. The main results are Theorems 9, 13, 15, 16, and 18.