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 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.

Mechanical decision for a class of integral inequalities

“…In this section, we review the Timefote's dimension-decreasing approach [7][8][9] and describe the mechanically decidable problems for a class of symmetric polynomial inequalities. A complete theoretical result is given.…”

“…Using Timofte's dimension-decreasing method for symmetric polynomial inequalities [7][8][9], combined with the inequality-proving package BOT-TEMA [1,[10][11][12] and a program which implements the method known as successive difference substitution [1,[13][14][15][16][17], we provide a procedure to decide the nonnegativity of the corresponding polynomial inequality such that the original integral inequality is mechanically decidable. For simplicity, all integrals mentioned in this paper are assumed to be Riemann's [18].…”

“…In [37], a class of symmetric polynomial inequalities with degree less than 5 is successfully transformed into finitely many inequalities independent of n by using Timfote dimension-decreasing algorithm [7][8][9]. The nonnegativity of these polynomials can be decided by using the inequality-proving package BOTTEMA [1,[10][11][12].…”

“…In section 2, we review Timofte dimension-decreasing approach [7][8][9], describe the mechanically decidable problem for a class of symmetric polynomial inequalities and present a complete theoretical result. In section 3, we improve the efficiency of the algorithm by employing successive difference substitution [1,[13][14][15][16][17], and present a further discussion on alternative approaches to dimension-decreasing.…”

A class of mechanically decidable problems beyond Tarski’s model

“…Intuitively speaking, v(x) counts the distinct components of x, while v * (x) counts the non-zero distinct components only. [11,12] . (…”