Positive sextics and Schur's inequalities (Q1175716)

The paper studies symmetric ternary forms over the reals. Definitions of positive semi definite forms, copositive forms, even forms, extremal forms, etc. give rise to various closed convex cones \(P_{3,2d}\), \(P_{3,d}^ +\) and their subcones \(\text{Sym }P_{3,2d}\), \(\text{Sym }P_{3,d}^ +\). The authors discover new extremal forms in \(P_{3,6}\) and \(\text{Sym } P_{3,6}^ +\) which in turn give some remarkable new symmetric extremal inequalities as for example \[ \sum n^ 4(y-z)^ 2\geq {\textstyle {1\over 2}}(x-y)^ 2 (y-z)^ 2 (z-x)^ 2 \qquad (\text{all } x,y,z\in\mathbb{R}). \] One of the methods used to determine these extremal forms is a striking application of classical Euclidean geometry of the \(q\) point circle going back to Euler, Chapple and Feuerbach. This paper may be viewed as an excellent contribution to the general theory of polynomial inequalities and lays ground work for the determination of the cones \(\text{Sym } P_{3,6}\) and \(\text{Sym }P^ +_{3,6}\).