The birational composition of arbitrary quadratic form with binary quadratic form
Let <i>f</i>(<i>X</i>) and <i>g</i>(<i>Y</i>) be non-degenerate quadratic forms of dimensions <i>m</i> and <i>n</i> respectively over a field <i>K</i>, char<i>K</i> ≠ 2. Herein, the problem of the birational composition of <i>f</i>(<i>X</i>) and <i>g</i>(<i>Y</i>) is considered, namely, the condition is established when the product <i>f</i>(<i>X</i>)<i>g</i>(<i>Y</i>) is birationally equivalent over <i>K</i> to a quadratic form <i>h</i>(<i>Z</i>) over <i>K</i> of dimension <i>m</i> + <i>n</i>? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms <i>f</i>(<i>X</i>) and <i>g</i>(<i>Y</i>) over a field <i>K</i> when <i>m</i> = 2. The sufficient and necessary conditions for the existence of birational composition <i>h</i>(<i>Z</i>) for quadratic forms <i>f</i>(<i>X</i>) and <i>g</i>(<i>Y</i>) over a field <i>K</i> for <i>m</i> = 2 are obtained. The set of quadratic forms is described which can be considered as <i>h</i>(<i>Z</i>) in this case.