zbMATH — the first resource for mathematics

On positive definite biquadratic forms irreducible to sums of squares of bilinear forms. (English. Russian original) Zbl 0886.11022
Mosc. Univ. Math. Bull. 50, No. 2, 28-32 (1995); translation from Vestn. Mosk. Univ., Ser. I 1995, No. 2, 29-33 (1995).
The paper considers positive definite biquadratic forms \[ \sum_{i,j=1}^{n} \sum_{\alpha,\beta=1}^{\mu} f_{ij}^{\alpha \beta}\xi ^{i}\xi ^{j}\eta _{\alpha}\eta _{\beta}. \tag{1} \] For \(n>2\), \(\mu >2\), F. J. Terpstra [Math. Ann. 116, 166-180 (1938; Zbl 0019.35203)] constructed an example of a form (1) which cannot be represented as the sum of squares of bilinear forms. The present paper generalizes this result by applying Terpstra’s construction and proves that if \(n>2\), \(\mu >2\) then there exists an open set of such forms in the space of biquadratic forms.
11E76 Forms of degree higher than two
11E39 Bilinear and Hermitian forms