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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.
MSC:
 1.1e+77 Forms of degree higher than two 1.1e+40 Bilinear and Hermitian forms