zbMATH — the first resource for mathematics

Real even symmetric ternary forms. (English) Zbl 0970.11013
In real algebra it is an important issue to decide which positive semi-definite forms over the real numbers can be represented as sums of squares of forms. The author studies this question for the space \(S^e_{n,m}\) of \(n\)-ary forms that have degree \(m\), are even (all variables occur only with even degrees) and symmetric (invariance under permutations of the variables). The cones of positive semi-definite (or sum of squares) forms in \(S^e_{n,m}\) are denoted by \(PS^e_{n,m}\) (or \(\Sigma S^e_{n,m})\).
As a first step, tests are devised for \(S^e_{n,8}\) and \(S^e_{3,10}\) that decide whether a given form is positive semi-definite or not. The tests are used to determine subsets \(U\subseteq PS^e_{3,8}\) and \(V\subseteq PS^e_{3,10}\) that contain all extremal elements in the two cones. The set \(U\) consists entirely of sums of squares, and it follows that \(PS^e_{3,8}= \Sigma S^e_{n,8}\).
On the other hand, the inclusion \(\Sigma S^e_{3,10}\subseteq PS^e_{3,10}\) is shown to be proper. For all elements of \(V\) it is decided whether they are sums of squares or not, which results in various new families of positive semi-definite forms that are not sums of squares.

11E76 Forms of degree higher than two
11E10 Forms over real fields
13J30 Real algebra
Full Text: DOI
[1] Bottema, O.; Groenman, J.T., On some triangle inequalities, Univ. beograd. publ. elektrotehn. fak. ser. mat., 577-598, 11-20, (1977) · Zbl 0367.50005
[2] Choi, M.D.; Knebusch, M.; Lam, T.Y.; Reznick, B., Transversal zeros and positive semidefinite forms, Lecture notes in mathematics, (1981), Springer-Verlag Berlin/New York, p. 273-298
[3] Choi, M.D.; Lam, T.Y., Extremal positive semidefinite forms, Math. ann., 231, 1-18, (1977) · Zbl 0347.15009
[4] Choi, M.D.; Lam, T.Y.; Reznick, B., Real zeros of positive semidefinite forms, I, Math. Z., 171, 1-26, (1980) · Zbl 0415.10018
[5] Choi, M.D.; Lam, T.Y.; Reznick, B., Even symmetric sextics, Math. Z., 195, 559-580, (1987) · Zbl 0654.10024
[6] M. D. Choi, T. Y. Lam, and, B. Reznick, Symmetric quartic forms, unpublished manuscript, 1980.
[7] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1967), Cambridge Univ. Press Cambridge
[8] Hilbert, D., Über die darstellung definiter formen als summe von formenquadraten, Math. ann., 32, 342-350, (1888) · JFM 20.0198.02
[9] Motzkin, T.S., The arithmetic – geometric inequality, (), 205-224
[10] Rigby, J.F., A method of obtaining related triangle inequalities, with applications, Univ. beograd. publ. elektrotehn. fak. ser. mat., 412-460, 217-226, (1973) · Zbl 0265.50007
[11] Rigby, J.F., Quartic and sextic inequalities for the sides of triangles, and best possible inequalities, Univ. beograd. publ. elektrotehn. fak. ser. mat., 602-633, 195-202, (1978) · Zbl 0438.51016
[12] Robinson, R.M., Some definite polynomials which are not sums of squares of real polynomials, Selected questions of algebra and logic, (1973), Akad. Nauk. USSR Moscow, p. 264-282
[13] Ursell, H.D., Inequalities between sums of powers, Proc. London math. soc. (3), 9, 432-450, (1959) · Zbl 0090.01504
[14] Walker, R.J., Algebraic curves, (1962), Dover New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.