Geodesic
The birational composition of arbitrary quadratic form with binary quadratic form
Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2022), pp. 14-20.

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Let $\mathit{f}(X)$ and $\mathit{g}(Y)$ be non-degenerate quadratic forms of dimensions $m$ and $n$ respectively over a field $K$, $charK \neq 2$. Herein, the problem of the birational composition of $\mathit{f}(X)$ and $\mathit{g}(Y)$ is considered, namely, the condition is established when the product $\mathit{f}(X) ~\mathit{g}(Y)$ is birationally equivalent over $K$ to a quadratic form $\mathit{h}(Z)$ over $K$ of dimension $m + n$? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms $\mathit{f}(X)$ and $\mathit{g}(Y)$ over a field $K$ when $m = 2$. The sufficient and necessary conditions for the existence of birational composition $\mathit{h}(Z)$ for quadratic forms $\mathit{f}(X)$ and $\mathit{g}(Y)$ over a field $K$ for $m = 2$ are obtained. The set of quadratic forms is described which can be considered as $\mathit{h}(Z)$ in this case.
Mots-clés : quadratic form; birational equivalence; birational composition. 
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A. A. Bondarenko. The birational composition of arbitrary quadratic form with binary quadratic form. Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2022), pp. 14-20. http://geodesic.mathdoc.fr/item/BGUMI_2022_1_a1/

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