On the positivity of symmetric polynomial functions. Part III: Extremal polynomials of degree 4

Abstract

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.

Mechanical decision for a class of integral inequalities

“…By now, using Timofte dimension-decreasing approach [1,[7][8][9]37], the nonnegativity decision problem for symmetric form of degree m in n variables in S 0 n,m has been successfully transformed into a problem of verifying inequalities in 2l (or 2l + 1) variables, where l depends on the degrees of given symmetric form. l may be very large when the degree of given symmetric form is high.…”

“…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.…”